Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://new.math.msu.su/department/probab/ktv80.pdf
Äàòà èçìåíåíèÿ: Wed Jun 10 23:49:02 2015
Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:06:52 2016
Êîäèðîâêà:

..

-

X 3
80- - ..

2015

519.2 22 C 56

. X. . 3. / .., ... .: , 2015. 208 . 80- .., 1935 , .., . , , XIV . , , , , .

-: ..

60 90/16. . . 13. . -34. . 117312, , - , 11, . 11.

ISBN 978-5-9710-2380-7

c , 2015 c , 2015

80- - ..

- .. 1935 . , .. (19031987). . . .. 1935 1966 . 1966 1995 .. (19121995). 1996 , ... , .., .., .., .., .., .., .., .., .., .., .., .., .. . 2015 37 , : 18, 13, 2, 4. - ( , , ), : , , , , , . 2014 4 ( ). . . . 1. ( , , ). 2. ( , , , , , , , , ). 3. , , , , . 2014 97 ( 71 63 ). 1621 2003 , 100- ... .. (- ). 4

1000 . 15 . 916 . 2630 2012 , 100- ... - . .. , . . 188 , 42, 41. 170 , 31. 400 , . 12 2014 80 ... , : , , , , , , , . . .. .. (1974), .. (1994), (1996), . (2011). (1994). . . ( ), (2000). (2002). (2005). (19941998), (19891991), (19981999); (1985); , ; (1990), . , . .. , 60 30 . , 9 ( , 4 ). 2003 ( .. ). 10 , 8 300 . .. ( ..), . - .., (2003). Stochastics, Statistics, Financial Mathematics (, 1315 2014, 80- ..), Probability, Analysis and Geometry ( , 30 4 2014), Probability Theory and its Applications (, 5

, 1214 2015, 85- ..). , 260- .., .. . ( ), ( .. ..). , .., .. .., . . .. () , , , . .. c 2007 IWS (International Workshop on Simulation), ASMDA (Applied Stochastic Models and Data Analysis) SMTDA (Stochastic Modeling Techniques and Data Analysis) . , IWS-2015 ASMDA-2015. , . . 25 2015 , .., XIV . 54 (18 , 36 ) - , . . , , ... 2015 (1315 ) 20 29. 2014 (4 ) : ( , 1533 ( ), ), ( .. , - , - , ), (, , , , , , , -, , , .). 2015 21 , 13 . 2014 . , .. 2014 , ( ..), .. 6

( ..). . C 2014 .. Introduction to Markov Processes ; .. Risk Models Mathematics of finance and investment , .. Stochastic inventory and insurance models . 2014 .. ( .C. , .. , ... .. ... ..) . , - . C 10 . .. , - . 2015 .. 60 . . (.., .., .., .., .., ..) (.., ..). . 2013 .. .. . .. 2014 , .. 2015 . 2014 .. , 2015 .. . . () 1976 . 1997 ... 2015 7 : (. .), 2 , 3 , 1 . : , . , . , . .. Extremes . , () FP7 Risk Analysis, Ruin and Extremes ( 7

, ). () 1976 . ... 2015 7 : (. .), 1 , 5 . , . . 1993 , - .. - ( ), . 2014 .. c () . ( ), , . () 1991 . ... 2015 4 : (. .), 1 , 2 . (, , , , ). ( ), . 1995 , .., Markov Processes and Related Fields . , , . 20 , ( ). 3 20 ... cc .. ( ) , 2020 ( 2020 ), , . .. . . ( ). (http://www.math.msu.su/department/probab), (-) (http://istina.imec.msu.ru/organizations/department/275918/workers/). , . 8

.., ..

1
..2 , ..3

. , .

1

Zd {n1 , n2 , . . . , nd } ni , d- . () Zd , ., , [15, 14, 16, 2, 10, 12]. (.. ). , , [15, 14, 10, 12]. N , Zd , , , [13]. [13]. . t . ,
( 13-01-00653). , eka.antonenko@gmail.com, , - . .. . 3 , yarovaya@mech.math.msu.su, , - . .. .
2 1

9

, . Zd N . , . , . , . , [13] B RW/N/0/0. [1, 10] B RW/1/0/0, , . , . , B RW/1/0/0 ( ). , B RW/1/0/0, , , , [10]. B RW/N/0/0 c , x1 , x2 , . . . , xN , () A,
N

H = A +
i=1 p d p d

xi ,

xi Zd ,

(1)

T A : l (Z ) l (Z ), p [1, ], , x = x x , x = x (·) - , x . [13] , N xi i=1 A H , N . , , H N - 1. [17] , (1) , , . [17] . [13] , H lp (Zd ), p [1, ]. , T C = {z C : |z - a(0)| |a(0)|}, a(0) = 0 A0 , . , H , C . [13]. . 2 B RW/N/0/0. 3

10

.., ..

"" Zd (1). 4 . 5 " " , , ( ) H d 3, .

2

Zd µt (y ) t y Zd , t = 0 , x Zd , .. µ0 (y ) = x (y ). , Zd µt = yZd µt (y ). p(t, x, y ) . Ex µ0 (·) = x (·), mn (t, x, y ) := Ex µn (y ) t mn (t, x) := Ex µn , n N. t C A = (a(x, y ))x,yZd , a(x, y ) = a(x - y ) x y . , , A . a(z ), z Zd , a(0) < 0, a(z ) 0 z = 0 a(z ) a(-z ). , zZd a(z ) = 0 |z |2 a(z ) < ,
z

(2)

|z |

z . , , A , ..
k

z Zd z1 , z2 , . . . , zk Zd , z =

zi a(zi ) = 0
i=1

i = 1, 2, . . . , k . p(t, x, y ) ; , p(t, x, y ) a(x, y ) (., , [4, 10]). A p(t, x, y ):

G (x, y ) :=
0

e-t p(t, x, y ) dt,

0.

, G0 = G0 (0, 0) . (2) ( ) G0 = d = 1, 2 G0 < d 3 (., , [10]). bn , n 0, bn 0 n = 1, b1 0 n bn = 0. x1 , . . . , xN f (u) = bn un , n=0 , r = f (r) (1) < r N. 1 = f (1), , . , -(a(0) + b1 ) [11]. 11

B RW/1/0/0 [15, 1, 10, 12] , , , (1) N = 1. A l2 (Zd ), , x1 , x1 . , A A. [13] , , B RW/N/0/0. p(t, ·, y ) p(t) l2 (Zd ), t y , , [10, 13], p(t) l2 (Zd ) dp = Ap, dt A (Au)(z ) :=
z Z
d

p(0) = y ,

a(z - z )u(z ).

, m1 (t, ·, y ), m1 (t) l2 (Zd ), l2 (Zd ) dm1 = H m1 , dt m1 (0) = y . (3)

, l (Zd ), m1 (t) = m1 (t, ·) m1 (0) = 1. (., , [5]), t (3) . H (1) [13].

3

c , > c H . , : 1. d = 1 d = 2, c = 0; d 3, c = (G0 )-1 N = 1 0 < c < (G0 )-1 N 2. 1. 0 > 0, (c , c + 0 ) H ( ) ( ), ( ) 0 c , c . , . . 12

.., ..

2. c . , . 3. N 2. > c H N , 0 ( ) > 1 ( ) · · ·
N -1

( ) > 0,

0 ( ) . , c1 , (c , c1 ) 0 ( ). 0 ( ), 1 ( ), . . . , N -1 ( ) , [13, 6]: 4. H ,
N

Vi -
j =1

G (xi - xj )Vj = 0,
N i=1

i = 1, . . . , N

(4)

{Vi }

.

, 1 ( ), . . . ,
N -1

( )

H (.. ), ( ). , "" x1 , x2 , . . . , xN . , , . , " " , ci , i 2, i 3 . , , . , z , z . 5. a(z ) , G (z ) .

13

1. N 2 x1 , . . . , xN (.. ), , x1 = {1, x2 = {0, ... xi = {0, ... xN = {0, 0, . . . , 0, . . . , 0}, 1, . . . , 0, . . . , 0}, 0, . . . , 1, . . . , 0}, 0, . . . , 0, . . . , 1}. (4) = 0. (5)

: 1 G (x1 , xN ) G (x1 , x1 ) - · · · G (x2 , x1 ) ··· G (x2 , xN ) det ... ... ... 1 G (xN , x1 ) · · · G (xN , xN ) -

, G (xi , xj ) = G (0, xi - xj ) = G (0, xj - xi ) = G (xj - xi ). G (u, v ) G (u - v ) , G (xj - xi ) i = j , , G (x1 - x2 ) = G (z ), z = x1 - x2 , .. G (xj - xi ) G (x1 - x2 ) = G (z ), G (xi , xi ) G (xi , xi ) G (xi - xi ) = G (0) = G , , (5) G G det · G : 1 - · · · G (z ) (z ) · · · G (z ) = 0, ·· ··· ··· 1 (z ) · · · G - i. i = j. (6)

C : 1 G - + (N - 1)G (z ) · · · G (z ) N -1 1 0 ··· 0 = 0, G - G (z ) - det ··· ··· ··· 0 ··· -1 G + (N - 1)G (z ) - 1 G - G (z ) - 1
N -1

= 0.

, c c1 : c = (G0 + (N - 1)G0 (z ))-1 , 14 c1 = (G0 - G0 (z ))-1 . (7)

.., ..

. 1 c1 (6) (7), |z | z , N , .. c1 = c (|z |) > 0. c , N , .. c = c (|z |, N ), , c (|z |, N ) 0, N z . , c (|z |, N ) 0 d = 1, 2.

4

1 2 . 1. > 0 H , µi () = 1, i = 1, ..., N , µi () G (x1 , x1 ) . . . G (x1 , xN ) G (x , x ) . . . G (x2 , xN ) G() = 2 1 (8) ............................ . G (xN , x1 ) . . . G (xN , xN ) 1. > 0 H = A + N xi , h = 0 i=1 :
N

Ah +
i=1

xi h = h.

R = (A - I ) :

-1

A R
N

h+
i=1

R xi h = 0.

x h = x (x , h),
N

h+
i=1

(xi , h)R xi = 0.

xk :
n

(xk , h) +
i=1

(xi , h)(xk , R xi ) = 0, k = 1, ..., n,

, Uk = (xk , h), :
n

Uk +
i=1

Ui (xk , R xi ) = 0,

k = 1, . . . , n.

(9)

, h , . , (y , R x ) = - 1 (2 )d 15 ei(,y-x) d, - ()

[- , ]

d

() = zZd a(z )ei(,z) [- , ]d a(z ). : ei(,y-x) 1 d, G (x, y ) := e-t p(t, x, y )dt = (2 )d [-,]d - () 0 , (y , R x ) = -G (x, y ). , (9) : G (x1 , x1 ) - 1 . . . G (x1 , xN ) G (x2 , x1 ) ... G (x2 , xN ) det ...................................... = 0 G (xN , x1 ) . . . G (xN , xN ) - 1 , = 0, det G() - 1 I = 0. (10)

, µi (), i = 1, ..., N , G(), , (10) , : µi () = 1, . 1. [10] ei(,y-x) 1 d. (11) G (x, y ) = (2 )d [-,]d - () () c 2 |()| C 2 c C [10]. , G (x, y ) 0 [10] rd-1 dr 0. r2 d 3 d = 1 d = 2. , G (x, y ) 0, G() , G() : µ0 () 0. , > 0 µ0 () = 1 ( ), c , c = 0. G0 (0, 0) < , G0 (x, y ) < x y . , G(0) < , , G() G(0) 0. µ < , µ0 () µ < . µ0 () = 1, , ( ) 0. 1 H , .. c > 0. c d 3. , H . 1 µ0 () = 1, 16 i = 1, . . . , N .

.., ..

1 . (12) µ0 () - [3] µ0 () G() x() . G() = G() = G (0, 0)I + B (), B () = G() - G (0, 0)I , , B () , N 2. x() : 0 = G()x() - µ0 ()x() = (G (0, 0) - µ0 ())x() + B ()x(). = 0 0 = G(0)x(0) - µ0 (0)x(0) = (G0 (0, 0) - µ0 (0))x(0) + B (0)x(0). x(0) , B (0)x(0). , G0 (0, 0) - µ0 (0) < 0. , (12) 1 1 1 < = . µ0 (0) G0 (0, 0) G0 N = 1 c c G0 - 1 = 0 1 c = G 0 . c = 3. 1 H µi () = 1, i = 0, 1, . . . , N - 1. µi () G(), , [8], , , . µi () ( ): 0µ
N -1

() . . . µ1 () µ0 ().

(13)

(11) G() , , µi () 0 i = 0, 1, . . . , N - 1. [6, . 2, . 6.8] µi () - 0. (11), , G() > 0, - [3] , µ0 () G() , .. (13) : 0µ
N -1

() . . . µ1 () < µ0 ().

(14)

(11) , G() > 0. 0 , : 17

1. 0 G() G(0) . µi () , . 1, 1 µ0 (0) = c µ1 (0) = 1 , , c1 [9, . 592]. d 3.

. 1: µi () , lim

µ0 () < .

2. 0 G() . d = 1 d = 2. G() 0, , µ0 () 0 c = 0. , µ1 () 0, .. c1 = 0. , 2.1.1 [10], , µ1 () 0 , .. c1 > 0. , . 2. (11), , G() , ( G()) [7], µ0 () G() . , 1 i H (. . 1) µi (i ) = 1 , i = 0, 1, . . . , N - 1. (15)

µi () 0, i ( ) N . (14) , H , 0 = 0 ( ), 1 µ0 (0 ) = , (16) 18

.., ..

. 2: µi () , lim

µ0 () = , lim

µ1 () < .

. 0 (16) 0 = 0. c . µ0 () , 0 = 0 ( ) . . 5. , G (z ) [10] ei(,z) 1 d, G (z ) = (2 )d [-,]d - () () a(z ), () = zZd a(z )ei(,z) , [- , ]d . , G (z ) = G (Rz ) R (.. , , ). (R) =
z Z
d

a(z )ei

(R,z )

=
z Z
d

a(z )ei

(,R z )

= a((R )-1 z )e
i(,z )

=
z Zd

=
z Z
d

a(z )e

i(,z )

= (),

a((R )-1 z ) = a(z ) z Zd , a(z ) z . , () . , 1 G (Rz ) = (2 )d ei(,Rz) 1 d = - () (2 ) ei(R ,z) d = - ()
d [- , ]
d

[- , ]

d

d

[- , ]

d

=

1 (2 )

ei(,z) d = G (z ) - ((R )-1 )

R, () = ((R )-1 ) [- , ]d , , , () 19

. , G (z ) .

5

d 3, c ci , i = 1, . . . , N - 1, . , , z1 , z2 , z3 z4 . 2. , - Zd , . z2 - z1 , z3 - z2 ,

. 3: . z4 - z3 z1 - z4 . 5 , G (z2 - z1 ) = G (z3 - z2 ) = G (z4 - z3 ) = G (z1 - z4 ). 1 a = G - , b = G (z2 - z1 ) = G (z3 - z2 ) = G (z4 - z3 ) = G (z1 - z4 ), c = G (z3 - z1 ) d = G (z4 - z2 ). (10) abcb b a b d det c b a b = 0. bdba 1 2 16b + c2 - 2cd + d2 + c + d = 0, a+ 2 a - c = 0, a - d = 0, 1 2 a- 16b + c2 - 2cd + d2 + c + d = 0, 2 0 , . . . , 3 H . = 0 , ~ G = 16b2 + c2 - 2cd + d2 + c + d = = 16(G0 (z2 - z1 ))2 + (G0 (z3 - z1 ))2 - 2G0 (z3 - z1 ) · G0 (z4 - z2 ) + (G0 (z4 - z2 ))2 + + G0 (z3 - z1 ) + G0 (z4 - z2 ). 20

.., ..

, e c 1~ c = (G0 + G)-1 . 2 : ~ (G0 - G0 (z3 - z1 ))-1 , (G0 - G0 (z4 - z2 ))-1 , (G0 - 1 G)-1 , 2 z1 , z2 , z3 z4 . 3. z1 , z2 , z3 z4 , - , . z3 - z1 z4 - z2

. 4: . z4 - z1 , G (z4 -z2 ) = b, G (z2 -z1 ) = G (z3 -z4 ) = c (10) a c det b d , z2 - z1 z3 - z4 , z3 - z2 1 , G - = a, G (z3 - z1 ) = , G (z3 -z2 ) = G (z4 -z1 ) = d. c a d b b d a c d b = 0. c a

a a a a + + - - b b b b + - + - c c c c + - - + d d d d = = = = 0, 0, 0, 0,

0 , . . . , 3 H . = 0 , c c = (G0 + G0 (z3 - z1 ) + G0 (z2 - z1 ) + G0 (z4 - z1 ))-1 . (G0 + G0 (z3 - z1 ) - G0 (z2 - z1 ) - G0 (z3 - z2 ))-1 , (G0 - G0 (z3 - z1 ) + G0 (z2 - z1 ) - G0 (z3 - z2 ))-1 , (G0 - G0 (z3 - z1 ) - G0 (z2 - z1 ) + G0 (z3 - z2 ))-1 , z1 , z2 , z3 z4 . 21

[1] .., .., Zd // , 1998, . 53, 5, . 229230. [2] .., .., // . ., 2004, . 49, 3, . 463484. [3] .., . .: , 1967. [4] .., .., . .: , 1973. [5] .., M.., . .: , 1970. 534 . [6] ., . .: , 1972. [7] .., . .: , 1962. [8] .., .., // , 2012, . 447, 3, . 265268. [9] ., ., . .: , 1989. [10] .., . .: - , 2007. ISBN 978-5-211-05431-8. [11] .., // , 2009, . 4, 1, . 119 136. [12] .., // . ., 2010, . 55, 4, . 705731. [13] .., // , 2012, . 92, 1, . 124140. [14] Albeverio S., Bogachev L.V., Branching random walk in a catalytic medium. I. Basic equations // Positivity, 2000, v. 4, 1, p. 41100. [15] Albeverio S., Bogachev L.V., Yarovaya E.B., Asymptotics of branching symmetric random walk on the lattice with a single source // C. R. Acad. Sci. Paris S´ I Math., 1998, v. er. 326, 8, p. 975980. [16] Vatutin V.A., Topchi V.A., Yarovaya E.B., Catalytic branching random walks and i queueing systems with a random number of independent servers // Teor. v¯ Mat. Imo ir. Stat., 2003, 69, p. 115. [17] Yarovaya E.B., Branching Random Walks with Several Sources // Mathematical Population Studies, 2013, v. 20, 1, p. 1426.

22

..

1
..2

. . , , , , . , . . .

1

. A(t) {n } (0 = 0), , n=1 {A(t) - A(n ), t n } {A(t), t 0}. (., , [12]), {n } . n=1 . -, , . , A(t) () (., , [21]), A (t) t A(t) = A ((t)), (t) = 0 (y )dy , , (t) [12]. , , [16], [15] . , [15] , . ,
13-01-00653. , afanas@mech.math.msu.su, , , - . .. .
2 1

23

. -, , , . , , . , , . . [9], [26]. , , , . . , .

2

. .

,

{A(t), t 0}, (A(0) = 0) R+ , (, F , P) , . , (, F , P) {F t , t 0}, A(t) . 1. {A(t), t 0} , {j }j =1 (0 = 0) {F t , t 0}, , {j }
j =1

= {j - j -1 , A(j

-1

+ t) - A(j -1 ), t [0, j - j -1 )}

j =1

. j , j j = j -j -1 (j = 1, 2, . . . ) . A(t) , , [0, t). , A(t) , j = A(j ) - A(j -1 + 0). , a a = Ej < µ = Ej < . 1 = limt A(t) = µ t A(t). , {j } , j =1 0 = 0 . , 1 {j } , , j =2 . , P{1 < } = 1. , , 1. , [25] [24], , . 24

..

. [14]. . 1. {i = 1, 2, . . . }, , P{1 = i} > 0 . , {j } . j =1 , j j . tn = inf {t 0 : X (t) n} n = tn - tn-1 (t0 = 0). 1. 1, {n+k } k=1 ^ ^ n {k } Ek = k=1 -1 . [14]. 2. > 0 E
2+

< , E

2+

< ,

(1)

tT -tT ^ AT (t) = A( )T C - T [0, h] A . 2 a2 2 2acov( , ) + 3 - , µ µ µ2

2 A = 2 2 = D , = D .

(2)

, ([9] . 1 §11) , (. [14]). , , , . 3. , A(t) p (0 < p < 1) 1 - p. , , Ap (t) . , , . A(t). {n } , n=1 (...) . n n- , .. Z (t) , t, Rd . , t 0, s [0, t) + Z (t) = (Z (s), t - s, {A(u) - A(s), s 25 u t}, A
(s)+1

, . . . , A(t) ),

(3)

(·) . (·) , Z (s) = 0, . Z (t) {nk } , Z (n - 0) = 0. , k=1 Zn = Z (n - 0) . R/G/r/ , r (F C F S ). {k = (1k , . . . , rk )} , ik k - , k=1 i- . Z (t) W (t) = (W1 (t), . . . , Wr (t)). Wi (t) i- t , t. , W (t) (3) (·). Q(t), t, . , , Z (t) = (Q(t), x1 (t), . . . , xr (t)), xi (t) i- , t. Z (t) , xi (t) = 0, Q(t) < r i- . , , , . . r , . . Q(t) Z (t), Y (t) = X (t) - Q(t) , (0, t), .. ~ . , Z (t) = (Q(t), Z (t)) ~ (3) (·). , , . 4. , Z (0)
t

lim P {Z (s) A} = P(A)

d P(A) R+ , P ({0}) > 0. Y (t) . {nk } , Z (nk - 0) = 0. k=1

[5].

3

, Z (t) = (Z1 (t), . . . Zd (t)), (3). 26

..

, z Rd Z (t), + > 0 - (z ) = {x : ||x - z || < } t , P {Z (t ) (z )|Z (0) = 0} > 0. B0 (Z (t)) , Z (t). , Z (t) (), Z (0) B0 (Z (t))
t

lim P {Z (t)

x} = F (x), x Rd , +

, F (x) Z (0). , , ( , ..), , .. . (., , [10], [16], [28]). . R/G/1/. {i } i=1 ....., , b = E i . W (t) , Q(t) t. wn = W (tn - 0), qn = Q(tn - 0). 1 G/G/1/ ([23],[10]) lim P {wn x} = (x) lim P {qn x} = F (x).
n n

, = b < 1. W (t) Q(t) . 2. 1 . 3. P {1 = 0} + P {1 = 1, 1 - t1 > 1 } > 0. [12] lim P {W (t) x} lim P {Q(t) x}. , , t t = b < 1. Z (t) = (Z1 (t), . . . , Zd (t)), (3). 2. {Z (t) = (Z1 (t), . . . , Zd (t)) , t 0} Rd , > 0 y < , + P
d i=1

Zi (t) < y > 1 - t

0.

3. Z (t) , p Zi (t) t , i = 1, d. 27

, Z (t) . , . Z (t), (3),
d

Zn = (Zn1 , . . . , Znd ) z (t) =
j =1

Zj (t), zn =

n j =1

Znj .

(·). 4. P {zn+1 = 0|zn = 0} > 0. 5. x < m(x) (x) > 0 , Z =
d

(Z1 , . . . , Zd ) B0 (Z (t))
j =1

Zj

x P zn

+m(x)

= 0|Zn = Z

>

(x). , Zn , . d 6. > 0 x R+ > 0 n0 , | P Zn x|Z0 = y - P {Zn x|Z0 = y } | < , ||y - y || < , n > n0 y , y B0 (Zn ). , [5]. 1. 4 5. 1. Z0 = y B0 (Zn ) Zn , 6 . 2. Zn Z0 = y B0 (Zn ), . Z (t) z (t) 2. 1, . R/G/r/ , . 2. (i) = {ik } i-1 , k=1 (i = 1, . . . , r). . 2 W (t) = (W1 (t), . . . , Wr (t)) Q(t), Wn = W (n - 0) Qn = Q(n - 0). [5] . 2. 1. 3
r -1

=
i=1

i

< 1,

(4)

Wn Qn . , , 2, W (t) Q(t) . 2. > 1, = 1
2+ E 2 < , E 2+ 1

< , E

2+ i1

< , i = 1, r

(5)

> 0, . 28

..

, (5) 1 ([9] .IV, §7) . [5] , 3 4 5, 1. , Qn , , , < 1. > 1, [22] . = 1 (5) lim P Q(t) t x = 2
x 0

t

e- 2 dy ,

y2

2 2 2 2 = A + r=1 i i3 i = Di1 . i 1 . , 4 [5], 1 , . 1 . [1] , , , [7], . , , [20], 1 , . [5].

4

. , , , . , , . . [17], [16]. [28]. [29]. {Z (t), t 0} (UZ , F (UZ )). {n }n=1 , n = n - n-1 n- (n = 1, 2, . . . , 0 = 0), {n } n=1 {n } , 1 n=2 29

. F (t) = P{2 t}, F1 (t) = P{1 t}, µ = E2 ¯ F (t) = 1 - F (t). , F (t) , , {j } . j =2 P(A, t) = P{Z (t) A}, A F (UZ ), t 0. [12], µ < F1 () = 1, limt P(A, t) = P(A) {P(A), A F (UZ )} . , (t) = supAF (UZ ) | P(A, t) - P(A)|. . 3. F (t) C1 e
-1 t

1 - F1 (t)

, 1 - F (t)

C e-t , t

0

(6)

1 > 0, > 0, C1 < , C < . > 0, C < , (t) C e- t . (7)

, F1 (0) = 1 P(A, t) h(t) {j } . [8] h(t) = µ-1 + r(t), |r(t)| j =1 C e-t > 0, C < . t

. V (t) =

F (y )dy .

4. F (t) : 1. lim
V (t) t V (2t)

< ;
F (t) t V (t)

2. F (t) 3. F1 (t) (t)

C (1 + t)- lim

= 0 > 2 C < ;

C1 (1 + t)- t C (1 + t)-
min(-1, )

0, > 0. t 0 C < .

[6]. Z (t), (3). , kj j =1 , Z (kj ) = 0. , , Z (t) , . F (t) . (., , [16], [8], [10]). GI /G/1. un
(1) n=1

un

(2)

n=1 U2 (x

..... U1 (x) ) . , . W (t) (1) (1) tn = u1 + . . . + un n- , , 30

..

t1 = 0. [23], W (t) , = (t, x) = P {W (t)
t x0

Eu Eu

x}, (x) = lim (t, x) (t) = sup |(t, x) - (t)|.

(2) 1 (1) 1

< 1.

W (t) {tn } , W (tn - 0) = 0, 3 4 n=1 (t). U1 (x) U2 (x) (6), [13], 3 , . ( i) , i > 1 0 < c1 , (i) c2 < (i) (i) c2 c1 U i (x) , i = 1, 2. (8) (1 + x)i (1 + x)i
i( , , U i (x) Lxx) x , i Li (x) . , , (8) 4, 42.2, 43.3 [10].

1. GI /G/1 (8) < 1. (t) t c(1 + t)1
-min(1 ,2 )

(9)

0 c < .

. , W (0) = 0. , , T , .. = min {n 1 : W (tn - 0) = 0} , T = u1 + . . . + u(1) . C (1 + x)-

2

(1)

43.3 [10] P { > x} 42.2 [10] P {T > x} T u1 T
(1) (2)

C < ,

CT (1 + x)

- min(1 ,2 )

.

(10)

u1 1, (8) P {T > x} ~ CT (1 + x)
- min(1 ,2 )

,

(11)

~ CT > 0. (10) (11) 4, (9). (t) , ui (x), (i = 1, 2) (8), , [10]. , (t) C1 (1 + t)1-j , ui (x) , uj (x), (i = j ; i = 1, 2; j = 1, 2) (8). , . 31

5

. , , ( , , ..). , , , , , ... (., , [11], [18], [19], [27]). GI /G/1, (., , [13]). .. [8], . , {un } {n } n=1 n=1 . E(n - un ) < 0, n- wn limn P{wn x} = (x).

D(x) = P{n - un d(s) = Ee
s(n -un )

x}, G(x) =
x

[1 - D(y )]dy ,

(s R) s+ = sup{s : d(s) < }.

. 5. ( [8], §21, 11). s+ > 0 d(s+ ) > 1. x 1 - (x) = C1 e C
1 -q x

(1 + o(1)),

(12)

, q

d(q ) = 1. (13)

. 6. ( [8], §21, 12). s+ = 0, G(x) (. [8, §21]), x 1 - (x) = 1 G(x)(1 + o(1)). E(un - n ) (14)

R/G/1/ , . 3, , . , 5 ( ), {un }. , ([19], 3.1). 1 wn , qn , Qn W (t), Q(t) [3]. , W (t) , Q(t) t, wn , qn (Wn , Qn ) tn (n ). , (x) t n , b(s) = Ee-s , G(z , s) = Ez e-s . 32

..

7. 1, 0 = sup{s : G(b(-s), s) < } > 0 G(b(-0 ), 0 ) > 1.
x

(15)

lim x

-1

ln[1 - (x)] = -q ,

(16)

q

G(b(-q ), q ) = 1. (17)

(16) wn 1 3.1 [19]. ([13], . 2, . XI, §6, 2). (16) - Wn , Wn (. [2]), 5, . . qn wn qn (. [8], §25). 7, (x) x , , . A(t).
M

(t, w) =
j =1

j 1(U (t) = j ).

(18)

1(A) A, U (t) M j 0, j = 1, M . Un U (t) n- , {Un } n=1 {Pij , i, j = 1, M , Pii = 0} µ-1 j U (t) j . A(t) U (t) {1}. {j , j = 1, M } U (t), = M j j . j =1 Ti U (t) {i} {1}(i = 1), xi , . Gi (z , s) = Ez xi e-sTi µi Gi (s, z ) = s + µi + i (1 - z ) G1 (z , s) = 1. G(s, z ) = µ1 s + µ1 + 1 (1 - z ) 33
M M

Pij Gj (z , s) i = 2, . . . M ,
j =1

(19)

P1j Gj (z , s).
j =2

(20)

(19) (20), , s > 0 Z1 (s) > 1 G(Z1 (s), s) = 1 Z1 (0) = 1. , 1 = sup{s : b(-s) < } > 0 1 < . , q b(-q ) = Z1 (q ), .

6

, . , . , , . , , , , . , . . , , . . (. [14], [5], [15], [4]). ^ AT (t) (. 2) , (. [30] ).

[1] .., // , 2015 ( ) [2] .., // . , 2005, . 41, 1, . 5469. [3] .., .., // , 2015 ( ) [4] .., .., ( ) // , 2008, . 44, 4, . 81100. 34

..

[5] .., .., // , 2013, . 58, 2, . 210-234. [6] ., // , 1986, . XXVI, . 602606. [7] .., // , 2011, . 56, 1, .145152. [8] .., . .: , 1972. [9] .., . .: , 1980. [10] .., . .: , 1999. [11] .., .., // , 2005, . 8, 2, . 69136. [12] .., .., . .: , 1967. [13] . . .: , 1984. [14] Afanasyeva L.G., Bashtova E.E., Coupling method for asymptotic analysis of queues with regenerative input and unreliable server // Queueing Systems, 2014, v. 76, 2, p. 125147. [15] Afanasyeva L.G., Bashtova E.E., Bulinskaya E.V., Limit theorems for semi-Markov queues and their applications // Communications in Statistics. Simulation and Computation, 2012, v. 41, 6, p. 688-709. [16] Asmussen S., Applied Probability and Queues. Springer, 2003. [17] Asmussen S., Kluppelberg C., Sigman R., Sampling at subexponential times, with queueing applications // Stochastic Processes and their Applications, 1999, v. 79, p. 265286. [18] Bryc W., Denibo A., Large Deviations and Strong Mixing // Ann. Prob. and Stat., 1996, v. 32, p. 549569. [19] Gantsh A., O'Connel N., Wischik D., Big Queues. Springer, 2004. [20] Gaver D.P., A waiting line with interrupted service including priority // J. Rl. Stat. Soc. 1962, v. 24, p. 73 90. [21] Grandel l J., Double stochastic Poisson processes. Lect. Notes. Math., 1976, v. 529. [22] Iglehart D.L., Whitt W., Multiple channel queues in heavy traffic. I // Advances in Applied Probability, 1970, v. 2, 1, p.150177. [23] Loynes R., The stability of a queue witk non-independent inter-arrival and service times // Proc. Camb. Phil. Soc., 1982, v. 58, 3, p. 497520. 35

[24] Morozov E., The stability of non-homogeneuos queueing system with regenerative input // J. Math. Sci., 1997, v. 46, p. 407421. [25] Morozov E., The tightness in the ergodic analysis of regenerative queueing processes // Queueing Systems, 1997, v. 27, 12, 179-203. [26] Morozov E., Weak Regeneration in Modeling of Queueing Processes // Queueing Systems, 2004, v. 46, 34, p. 295315. [27] Sadowsky J.S., Szpankowski W., The Probability of Large Queue Length and Waiting Times in a Heterogeneous Multiserver Queue. Part I: Tight Limits // Adv. Appl. Prob., 1995, v. 27, p. 532566. [28] Thorisson H., Coupling, stationarity, and regeneration. Springer, 2000. [29] Veretenikov A.Yu., On the rate of convergence for infinite server Erlang-Sevastyanov's problem // Queueing Systems, 2014, v. 76, 2, p. 181203. [30] Whitt W., Stochastic-process limits: an introduction to stochastic-process limits and their application to queues // Springer Science & Business Media, 2002.

36

.., ..

1
..2 , ..
3

/ . , . , . , , . .

1

(., , [3], [7]), , , , , , , , , , . -, (T , Z, Y , U, , L). T , Z , Y U . , X X = (Z, Y , U ). L , . . 1951 [10] 1955 [14]. [11]. , , . [13]. , (.. )
13-01-00653. , ebulinsk@yandex.ru, , - . .. . 3 , alun@mail.ru, , - . .. .
2 1

37

. , . . , , . [18] [19]. .. [21], [22] .. [16], , 1962 . , , (., , [1], [2], [8]). , , , , . . [20]. , (., , [9], ). , / . 2 . 3 . , , . 4 . , 5 .

2

. 1. Tn , n 1, F (t) = P(Tn t). S0 = 0, Sn = T1 + . . . + Tn . {Sn } 0 . Sn , n 0, . Tn ,

µ=
0

y dF (y ) =
0

[1 - F (y )]dy ,

0

µ . µ = µ-1 = 0. B [0, ) N (B ) Sn , B . Nt = N ([0, t]) [0, t], .. Nt = min{k 0 : Sk > t}.

{Nt > n} , n- [0, t], P{Nt > n} = P{Sn t}. 38

.., ..

U (t) = ENt =
n=0

P{Nt > n} =
n=0

P{Sn

t} =
n=0

F n (t),

[0, t]. F n (t) n- F (t) , F 0 (t) F 0 (t) = 0, t < 0, 1, t 0.

U (t) . , . , , [1], [2], [8]. 1. U (t) t [0, ). 1 ( ). t t-1 U (t), µ-1 . Z (t) = z (t) + (F Z )(t), Z = z + F Z . 1. , . 2. z (t) , t < 0, t

Z = U z , Z (t) =
0

z (t - y )dU (y ),

Z = z + F Z , , (-, 0) . 2 ( ). F , z , Z Z = z + F Z

1 lim Z (t) = t µ
0

z (y )dy .

3 ( ). 0 < ET1 = µ < , t Nt p 1 , t µ , , . 39
p

4 ( ). ET1 = µ, D T1 = 2 < , t Nt - tµ t
d -1 2 µ-3

N (0, 1).

d

, , N (0, 1) , 0 1. 2. T0 , c {Tn } , 1 , G0 (t) = P(T0 t), F (t) , {Sn } , Sn = T0 + T1 + . . . + Tn , 0 .

3
3.1

. 3. Sn = n=0 Xi , 0 n k - 1, n k i Sn = Sk-1 + T1 + . . . + Tn-k+1 , , Gi (x) (..) Xi , i = 0, . . . , k - 1. Tj , j 1, .. F (x). Xi , i 1, Tj , j 1, , Sn , n 0, . k > 1 {Sn } 0 . 4. S0 = 0, Sn = T1 +. . .+Tn , n 1, Tj , j 1, , Fi (x) .. Tkl+i , i = 1, . . . , l, l > 1, k = 0, 1, 2, . . .. Tj , Sn , {Sn } 0 l. , l = 2. k 0 S2k , S2k+1 (). , [0, t]. 5. , Sn = n n k - 1, n k Sn = Sk-1 + T1 + . . . + Tn-k+1 , i=0 Xi , 0 , Gi (x) .. Xi , i = 0, . . . , k - 1. Fj .. Tql+j , j = 1, . . . , l, l > 1, q = 0, 1, 2, . . .. Xi , Tj , Sn . {Sn } 0 (k , l)-. . 40

.., ..

3.2

V (t) ,

V (t) =
n=0

P{S

n

t} =

= G0 + G0 G1 + . . . + G0 G1 . . . Gk

-1

+
r=1

G0 G1 . . . Gk

-1

F

r

=
-1

= G0 + G0 G1 + . . . + G0 G1 . . . Gk

U,

U (t) = F r (t), , . r=0 , . 5.
t

lim

U (t) V (t) = lim . t t t

. V (t) V = G0 + G0 G1 + . . . + G0 G1 . . . Gk
-2

+ G0 G1 . . . Gk

-1

U.

(1)

k - 1 X0 , X0 + X1 , . . . , X0 + X1 + . . . + Xk-2 , . U (t) X0 + X1 + . . . + Xk-1 , , U (t). , V (t) k - 1 + U (t) , t t , U (t) V (t) lim . (2) lim t t t t , U1 = g U, g X0 + X1 + . . . + Xk-1 . , g (x) , 0 < < 1
t-t

U1 (t)
0

g (t - y )dU (y )

g (t)U (t - t).

, U1 (t) t lim
t

U (t - t) (1 - )g (t), t - t U1 (t) t lim U (t) . t

t

V (t) U1 (t) k - 1 , V (t) U (t) lim lim , t t t t , (2), . 41

,
t

lim

V (t) = µ-1 , µ = ET1 . t

, . 6. F (x) µ 2 , Gi (x) i , i = 0, 1, . . . , k - 1, t V (t) - t 2 + µ 2 0 + 1 + . . . + - µ 2µ2 µ
k-1

+ k - 1.

(3)

. (1) V (t) , V = z1 + F V , z1 = V - F V = G0 + G0 G1 + . . . + G0 G1 . . . Gk-1 - F G0 - F G0 G1 - . . . - F G0 . . . Gk-2 . (4) Z1 (t) = V (t) - tµ-1 . Z1 (t) Z1 = z1 + F Z1 ~ , (4), z1 = Z1 - F Z1 = F ~
t t

t t + z1 - . µ µ

(5)

t F = µ
0

t-y t 1 dF (y ) = - µ µµ
0 t -1 0

1 - F (y ) dy .

(6)

(5) (6), , z1 = z1 - µ ~

(1 - F (y ))dy , 1 µ

1 - F (y ) dy dt.

z1 (t)dt = ~
0 0

z1 (t) - 1 dt +
0

t

z1 (t) - 1 dt =
0 0

(1 - F G0 ) + (1 - F G0 G1 ) + . . . + (1 - F G0 . . . Gk-2 )-

- (1 - G0 ) - (1 - G0 G1 ) - . . . - (1 - G0 . . . Gk-1 ) dt = = (µ + 0 ) + . . . + (µ + 0 + . . . + k-2 ) - 0 - (0 + 1 ) - . . . - (0 + . . . k-1 ) = = (k - 1)µ - (0 + . . . + k-1 ). (7) 42

.., ..

,

1 µ

1 - F (y ) dy

1 1 - F (y ) dy dt = t µ
t t

+
0 0

t 1 - F (t) dt = µ

0

1 = µ
0

1 t 1 - F (t) dt = 2µ
0

1 1 - F (t) dt2 = 2µ
0

t2 dF (t) =

2 + µ2 . (8) 2µ

, (7), (8)

z1 (t)dt = ~
0

2 + µ2 + (k - 1)µ - (0 + . . . + 2µ

k -1

).

,

1 |z1 | = z1 - 1 + ~ µ
t

1 - F (y ) dy

1 µ
t

1 - F (y ) dy + |z1 - 1|

1 µ
t

1 - F (y ) dy + (1 - F G0 ) + (1 - F G0 G1 ) + . . . + (1 - F G0 . . . Gk-2 ) +(1 - G0 ) + (1 - G0 G1 ) + . . . + (1 - G0 . . . Gk-1 ).

, , , , . , z1 , z1 ~ ~ (., , [1]). , (3). . , , . 7. F (x) µ < , t 1 Nt .. t µ . P(Nt > n) = P(Sn t),

Sn n 1. k , n-1 Sk-1 0 .. n . n- n-1 i=1k+1 Ti µ 1 . , n-1 Sn µ . 43

A = {|t-1 Nt - µ-1 | t A= , A = {Nt t r = [t(µ A
- t -1

}, t =
>0 t0 t>t
0

Nt t

1 , µ

A . t

t(µ

-1

- )} {Nt

t(µ

-1

+ ) } = A

- t

A+ . t

- )], [c] c, , r} = {Sr > t} Sr µ > r 1 - µ = Sr -µ> r
1 + = Br 1 ,

= {Nt

1 = µ2 (1 - µ). , > µ-1 P(A- ) = 0. < µ-1 , t r t . , A- =
>0 t0 t>t
0

A

- t

1 >0 r0 r >r0

+ Br 1 = B + .

l = [t(µ-1 + )] > t(µ-1 + ) - 1, , l t t (l + 1)(µ-1 + )-1 . , A
+ t

= {Nt

t(µ

-1

+ )} {Nt
-1

l} = {Sl < t}

S

l

l+1 µ +
-1

=

Sl -µ l

-

2

,

2 = (µ - l-1 )(µ

+ )

-1

> 0 l > (µ)-1 . , A+ t
>0 t0 t>t0
2

A+ = , ,

Bl-2 = B - .
l0 l>l
0

A = A- A+ B + B - = B = {n-1 Sn

µ, n }.

, P(B ) = 0, Nt 1 t µ .. n .

, . , . 8. ET1 = µ, D T1 = 2 > 0, EXi = i , i = 0, . . . , k - 1, t Nt - tµ-1 d N (0, 1). tµ-3 44

.., ..

. , Sn , , P Sn - ESn n v (v ) n .
-1/2

n ESn ) (t) = Z t n
k -1

Zn = ( n)-1 (Sn -
n-k+1

Z n n

n

=
j =0

Xj -j

t n

·

T1 -µ

t n

.

T1 , , .. t2 ¯ T1 (t) = T1 (0) + tT1 (0) + T1 (0) + o(t2 ). 2 , t2 2 ¯ T1 -µ (t) = 1 - + o(t2 ). 2 ¯ , Xj -j (t) = 1 + oj (t). ,
k-1

Z n n

(t) =
j =0

t2 ¯ ¯ (1 + oj (t)) · 1 - + o(t2 ) 2n

n-k+1

e- 2 .

t2

, k , k . , (Sn - ESn )( n)-1 , , . Nt P(Nt > n) = P(Sn P(Nt > n) = P P Sn - ESn n
k -1

t). t - ESn n

Sn - ES n

n

.

v

1 (v ) = 2

v

e-
-

x2 2

dx n .

m =
j =0

j . n t , t - ESn t - m - (n - k + 1)µ = = v. n n 45

v , n t-a + n= µ v v ± 2 tµ 2µ
2

1+

v 2 2 -4aµ 4tµ

v tµ t =± µ µ2

1+O ¯

1 t

,

a = m - (k - 1)µ. n, , P(Nt > n) = P Nt - tµ- tµ-
3 1

±v + O ¯

1 t

(v ).

, v . (v ) , Nt - tµ-1 P -v (v ) = 1 - (-v ) tµ-3 Nt - tµ-1 P v (v ). tµ-3

3.3

0

W (t). t W (t) =
n=0

P{Sn

t} = 1 + F1 + F1 F2 + . . . + F1 . . . Fl + F1 . . . Fl F1 + . . .
l -1

= 1 + F1 + F1 F2 + . . . + F1 . . . F P(S0 t) = 1 t.

+ F1 . . . Fl W,

W = z2 + H W, z2 = 1 + F1 + F1 F2 + . . . + F1 . . . F U2 (t) =
n=0 n l -1

(9)

, H = F1 . . . Fl .

H (t) =
n=0

(F1 . . . Fl )n (t).

(10)

2 , (9) W (t) = U2 z2 (t). . . 9.
t

lim

W (t) l = . t E(T1 + . . . + Tl ) 46

(11)

.., ..

. , W (t) = U2 z2 (t), z2 = 1 + F1 + F1 F2 + . . . + F1 . . . Fl-1 l ( 0, T1 , T1 + T2 , . . . , T1 + . . .+Tl-1 ), .. , l. , W (t) lU2 (t), a , W (t) lU2 (t) lim lim . t t t t . 0 < < 1
t-t

W (t)
0

z2 (t - y )dU2 (y )

z2 (t)U2 (t - t),

z2 (x) . , 0 < < 1 W (t) t U2 (t - t) (1 - )z2 (t). t - t

, , z2 (t) l t , lim
t

W (t) t

t

lim

lU2 (t) . t

, , U2 , (11). . 10. H (x) µ 2 , Fi (x) µi , i = 1, . . . , l, (µ1 + . . . + µl = µ), t 2 + µ2 (l - 1)µ1 + (l - 2)µ2 + . . . + µl lt - W (t) - l µ 2µ2 µ
-1

.

(12)

. Z2 (t) = W (t) - ltµ-1 . Z2 (t) Z2 = z2 + H Z2 ~ z2 (t) = Z2 (t) - H Z2 (t) = W (t) - ~
t

lt lt lt lt - H W (t) + H = z2 + H - , µ µ µ µ

lt l lt H = - µ µµ
0

1 - H (y ) dy .

, z2 = (z2 - l) + ~

l µ

1 - H (y ) dy ,
t

l µ

1 - H (y ) dy dt.

z2 (t)dt = ~
0 0

z2 (t) - l dt +
0

t

47

2+ 2 6, F H , l 2µµ . :

z2 (t) - l dt =
0 0

(1 + F1 + F1 F2 + . . . + F1 . . . Fl

-1

- l)dt =

=-
0

(1 - F1 ) + (1 - F1 F2 ) + . . . + (1 - F1 . . . F
l-1

l-1

) dt =
l-1

= - µ1 + (µ1 + µ2 ) + . . . + (µ1 + . . . + µ

) = - (l - 1)µ1 + (l - 2)µ2 + . . . + µ

. (13)

z2 (t)dt = ~
0

2 + µ2 - (l - 1)µ1 + (l - 2)µ2 + . . . + µ 2µ

l-1

.

,

l |z2 | = z2 - l + ~ µ
t

1 - H (y ) dy

l µ
t

1 - H (y ) dy + |z2 - l|

l µ
t

1 - H (y ) dy + 1 - F1 (t) + 1 - F1 F2 (t) + . . . + 1 - F1 . . . F

l-1

(t) .

, , , , . z2 , , z2 ~ ~ . , , (12) . (). 11. Fi (x) µi , i = 1, . . . , l, t Nt µ .., t l µ1 + . . . + µl = µ. . 7, Sn n 1. i = l =1 T(i-1)l+j , i 1. n = q l + r, q j nl-1 , 0 r < l.
q ^ Sn = Sq + r ,

^ Sq =

q i=1

q i , r =

r j =1

Tq

l+j

, 0. , Sn q 1^ 1q = · Sq + r . n nq n 48 (14)

.., ..

i , i 1, ^ µ = µ1 + . . . + µl q -1 Sq µ .. q ^ ^ Ti , r < q+1 = Sq+1 - Sq . n q , q n-1 l-1 . n (14), lim µ Sn = n l ...

n

, 7, , P Nt l t µ .. Nt l t µ P Sn µ n l = 1.

2 12. Fi (x) µi i , i = 1, . . . , l, t Nt - tlµ-1 d N (0, 1). l tµ-3 2 µ = µ1 + . . . + µl , 2 = 1 + . . . + l2 .

. , Sn Sn - ESn v (v ) n , P q n = lq + r, 0 r < l, .. q = n . l :
Zn q

Z n q

=

Sn -ESn q

-

(t) = Z

n

t

q

=

T1 +...+Tl -µ

t

q

r

q

·
i=1

Ti -µi

t

q

.

, , (t) = (0) + t (0) + t2 ¯ (0) + o(t2 ), 2

, , (t) = t2 ¯ 1- + o(t2 ) 2q
q r -1

Zn q

·
i=0

1-

t2 i 2q

2

¯ + o(t2 )

e

-

t2 2

.

, l , r l - 1 , q n . 49

-E , SnqSn .. , , . Nt

P(Nt > n) = P
r

Sn - ES q

n

t - ESn q

.

m =
j =1

µj . n t , t - ESn t - m - qµ = = v. q q

v , q . t - m v v ± 2 tµ 1 + + q= µ 2µ2
v 2 2 -4mµ 4tµ

v tµ t =± µ µ2

1+O ¯

1 t

.

n = q l + r, , P(Nt > n) = P Nt - tlµ-1 tµ-3 r tµ
-3

± lv + O ¯

1 t

=P

Nt - tlµ-1 tµ
-3

±l v + O ¯

1 t

(v ).

, v . (v ) , Nt - tlµ-1 -lv (v ) = 1 - (-v ) P tµ-3 Nt - tlµ-1 P v (v ). l tµ-3

3.4

[0, t], t . N0 (t) , [0, t], N1 (t) () , .. N0 (t) = min{k 0 : S2k > t}, N1 (t) = min{k 0 : S2k+1 > t}. Vi (t) = ENi (t), i = 0, 1. F0 (t) , F1 (t) . 50

.., ..

Vi (t) =

r=0

P{Ni (t) > r} =

k=0

P{S2

k +i

t}.

V0 (t) =
k=0

P{S2k

t} = 1 + F0 F1 +(F0 F1 )2 + . . . = U3 (t),

V1 (t) =
k=0

P{S2k

+1

t} = F0 + F0 (F1 F0 ) + F0 (F1 F0 )2 + . . . = F0 U3 (t),

U3 (t) =

(F0 F1 )n (t).
n=0

U3 c ( 0, T0 ), , . 13. V0 (t) V1 (t)
t

lim

V1 (t) U3 (t) V0 (t) = lim = lim . t t t t t

. V0 (t) , F = F0 F1 . V1 (t). , V1 (t) = F0 U3 (t), , V1 (t) U3 (t) U3 (t) V1 (t) lim . lim t t t t ,
t-t

V1 (t)
0

F0 (t - y )dU3 (y )

F0 (t)U3 (t - t),

F0 (x) . , V1 (t) U3 (t - t) (1 - )F0 (t), t t - t 0 < < 1. V1 (t) U3 (t) lim lim . t t t t ,
t

lim

V1 (t) U3 (t) = lim . t t t

, , U3 (t), V0 (t) V1 (t) 1 lim = lim = . t t t t E(T0 + T1 ) 51

2 14. Fi (x) µi i , i = 0, 1, t

V0 (t) -

2 + µ2 t , µ 2µ2

V1 (t) -

2 + µ2 µ0 t -, µ 2µ2 µ

2 2 µ0 + µ1 = µ, 2 = 1 + 2 .

. V1 (t) V1 = F0 + F V1 , F = F1 F0 .

t Z3 (t) = V1 (t) - µ . Z3 (t) Z3 = z3 + F Z3 , ~ t

t 1 t z3 (t) = Z3 (t) - F Z3 (t) = V0 (t) - - F V0 (t) + F = F0 (t) - 1 + ~ µ µ µ
0

1 - F (y ) dy ,

1 µ

1 - F (y ) dy dt.

z3 (t)dt = ~
0 0

F0 (t) - 1 dt +
0 +µ 2µ
2

t
2

, 6,

,

F0 (t) - 1 dt = -µ0 .
0

,

z3 (t)dt = ~
0

2 + µ2 - µ0 . 2µ

,

1 |z3 | = F0 - 1 + ~ µ
t

1 - F (y ) dy

1 µ
t

1 - F (y ) dy + (1 - F0 ).

, . z3 , , ~ z3 . ~ , ,

1 t V1 (t) - µ µ
0

z3 (t)dt = ~

2 + µ2 µ0 -, 2µ2 µ

t .

V0 F0 (t) 1. 52

.., ..

15. Fi (x) µi i = 0, 1, t 1 N0 (t) , t µ µ = µ0 + µ1 . . {Ni (t) > k } = {S2k
+i

N1 (t) 1 .., t µ

t} .

( 7) , n-1 S2n+i µ .., t-1 Ni (t) µ-1 .., i = 0, 1. n-1 S2n+i 11, . 16. Fi (x) µi 2 i , i = 0, 1, t Ni (t) - tµ-1
2 2 µ0 + µ1 = µ, 0 + 1 = 2 .

tµ-3

0 N (0, 1),

d

. P(Ni (t) > n) = P(S
2n+i

t).

, 12, Sn : Sn - ESn P v (v ) n , q n = 2q + r, 0 r 1, .. q = n . 2 , 12, .

3.5

W1 (t) .

W1 (t) =
n=0

P{S

n

t} = G0 + G0 G1 + . . . + G0 . . . Gk-1 + (F1 + F1 F2 + . . . + F1 F2 . . . Fl + F1 F2 . . . Fl F1 + . . .) = = G0 + G0 G1 + . . . + G0 . . . Gk-2 + G0 . . . Gk-1 W, (15)

+ G0 . . . Gk

-1

W (t) , . , H = F1 . . . Fl . 53

17.
t

lim

lU2 (t) W1 (t) = lim , t t t

U2 (t) (10). 5, 9, . , (11) lim l W1 (t) = . t E(T1 + . . . + Tl )

t

18. H (x) 2 µ 2 , Fi (x) µi i , i = 1, . . . , l, Gj (x) j , j = 0, 1, . . . , k - 1, t W1 (t) - 2 + µ2 (k - 1)µ - l - ((l - 1)µ1 + (l - 2)µ2 + . . . + µ lt l + µ 2µ2 µ
l-1

)

,

(16)

2 µ = µ1 + . . . + µl , = 0 + . . . + k-1 , 2 = 1 + . . . + l2 .

. W1 (t), W1 = z4 + H W1 , z4 = W1 - H W1 = G0 + . . . + G0 . . . Gk -H G0 + . . . - H G0 . . . Gk
-2 -2

+ G0 . . . Gk
-1

-1

U2 z2

- H G0 . . . Gk

U2 z2 =
-1

= (G0 + . . . + G0 . . . Gk-2 ) (1 - H ) + G0 . . . Gk

z2 .

, z2 = W - H W = 1 + F1 + F1 F2 + . . . + F1 . . . Fl-1 . Z4 (t) = W1 (t) - ltµ-1 . Z4 (t) Z4 = z4 + H Z4 ~ z4 (t) = Z4 (t) - H Z4 (t) = W1 (t) - ~
t

lt lt lt lt - H W1 (t) + H = z4 + H - µ µ µ µ

l = z4 - µ
0

l 1 - H (y ) dy = (z4 - l) + µ
t

1 - H (y ) dy ,

l µ

1 - H (y ) dy dt.

z4 (t)dt = ~
0 0

z4 (t) - l dt +
0

t

2+ 2 6, l 2µµ . 54

.., ..

z4 (t) - l dt =
0 0

(1 - H G0 ) + (1 - H G0 G1 ) + . . . + (1 - H G0 . . . Gk-2 )

- ((1 - G0 ) + . . . + (1 - G0 . . . Gk-2 )) - (l - G0 . . . Gk-1 z2 ) dt = (µ + 0 ) + (µ + 0 + 1 ) + . . . + (µ + 0 + . . . + k-2 ) - (0 + . . . + (0 + . . . + k-2 ) - ( + ( + µ1 ) + ( + µ1 + µ2 ) + . . . + ( + µ1 + . . . + µl-1 )) = (k - 1)µ - l - (l - 1)µ1 - . . . - µl-1 .

z4 (t)dt = l ~
0

2 + µ2 + (k - 1)µ - l + (l - 1)µ1 + . . . + µ 2µ

l -1

.

(17)

,

l |z4 | = z4 - l + ~ µ
t

1 - H (y ) dy

l µ
t

1 - H (y ) dy + |z4 - l|

l µ
t

1 - H (y ) dy + (1 - G0 (t)) + . . . + (1 - G0 . . . Gk-1 ) z2 )

+(1 - H G0 ) + . . . + (1 - H G0 . . . Gk-1 ) + (l - G0 . . . Gk

-1

l = µ
t

1 - H (y ) dy + (1 - G0 (t)) + . . . + (1 - G0 . . . Gk-1 ) +(1 - H G0 (t) + . . . + 1 - H G0 . . . Gk-1 )

+(1 - G0 . . . Gk

-1

F0 ) + . . . + (1 - G0 . . . Gk

-1

F0 . . . F

l -1

).

, , , , . z4 , , z4 ~ ~ . , , (17) , (16). 19. Fi (x) µi , i = 1, . . . , l, t Nt l .., (18) t µ µ = µ1 + . . . + µl . 55

. 1 n-1 Sn n . n > k q ^ Sn = Sk-1 + Sq + r , q ^ , 11, Sq = q=1 i , i = l =1 T(i-1)l+j , r = r=1 Tql+i , q = [(n - k + 1)/l]. i i j ,
q ^ Sn Sk-1 q Sq r = +· +. n n nq n

(19)

q n , i , i 1, ^ Ei = µ, q -1 Sq µ 1 . k , n-1 Sk-1 0 q r ^ ^ n .. , r q+1 = Sq+1 - Sq , n-1 q 0 n . , n-1 q l-1 n . , (19) n-1 Sn µ/l n . {Nt > n} = {Sn t}, , 7, (18).
2 20. Fi (x) µi i , i = 1, . . . , l, Gj (x) j , j = 0, . . . , k - 1, t

Nt - tlµ-1 l tµ-
3 2 µ = µ1 + . . . + µl , 2 = 1 + . . . + l2 .

N (0, 1),

d

. 8, Sn , P Sn - ESn q v (v ) n ,

n - k + 1 = lq + r, 0 r < l, .. q = n-k+1 . l 12 . Nt , P(Nt > n) = P Sn - ES q
k -1 n

t - ESn q

.

12 ,
r

m =
j =1

µj , =
j =0

i .

, n t , t - ESn t - a - qµ = =v q q a = m + , , n = k - 1 + lq + r, v . , t - a (v )2 v 4(t - a)µ + (v )2 t v tµ 1 q= + ± =± 1+O . 2 2 2 µ 2µ 2µ µ µ ¯ t n, . 56

.., ..

, [5] ( (k1 , k2 )-) Xi , .

4

, .. , renewal-reward processes. , . [2] .

4.1

6. (Ti , Ri ), i 1. Yt = Nt Ri i=1 . , , Nt = min{n : n=1 Ti > t}. i 3. µ = ET1 < , = ER1 < .
t

lim

Yt = t µ

..

(20)

. (20), Yt Nt 1 = · t t Nt
Nt

Ri ,
i=1

(21)

, t-1 Nt µ-1 t 1. Nt t n-1 n=1 Ri n , (21) i t 1, (20). 4. Yt , t h t 0, , .. EY
t+h

- E Yt h . µ

(22)

. , EYt = U (t), U (t) , {Ti } , = ER1 . U (t), 1 (22) . . 7. {Ti } l1 1 , {Ri } 1 l2 . Nt Yt = i=1 Ri . 57 -

, , S0 = 0, Sn = n=1 Ti , Nt = min{n : i Sn > t} [0, t]. , , .. Ri . µ = E l1 Tj l1 +i , = E l2 Rj l2 +i , j 0. , i=1 i=1 . 21. l1 Yt =· .. t . (23) lim t t l2 µ . , 3, .. Yt Nt 1 = · t t Nt
Nt

Ri .
i=1

11 1 t-1 Nt µ-1 l1 t . , Nt .. t . , , n-1 n=1 Ri . n = q l2 + r, 0 r < l2 , i
n q l
2

r

Ri =
i=1 j =1

j +

q r

,

j =
i=1

R(j

-1)l2 +i

,

q r

=
i=1

Rq

l2 +i

.

, j , , , q n . , , .. 1 - q q -1 q=1 j q , q n-1 l2 1 . , n-1 r j l2 l2 q ^ ^ 0 n . , |r | i=1 |Rq l2 +i | = q +1 . j = i=1 |R(j -1)l2 +i |, j 1, , , .. . , 1^ q n 1^ < q q q+1 1 = · q q+1
q +1

+1

+1

i=1

1 ^ j - q

q

^ j 0.
i=1

(23), . , Ri {Ti }, . 22.
t

lim

E Yt l1 = ·. t l2 µ

(24)

. Ri ,
[Nt /l2 ] [Nt /l2 ]+1

j
j =1

Yt <
j =1

j ,

(25)

58

.., ..

[a], a, j = l2 R(j -1)l2 +i . i=1 Nt {Ri } , , {j } , 1 1 , Yt (25). E[Nt /l2 ] = Ej , j 1. , Nt Nt Nt -1< < +1 l2 l2 l2 lim ENt E Yt = · lim . t t t l2 Nt + 1, l2 EYt < E([Nt /l2 ] + 1),

t

9 (24).

4.2

. - . 1. , c, {Ti } l1 . {Ri } l2 . Zt t Zt = ct - Yt , , . , . 21 1 lim l1 Zt =c- · , t l2 µ

t

22 limt t-1 EZt , .. . l1 , c - l2 · µ > 0, , Zt t . , , , . (0, t) [0, t], . , Ri , i = 1, 2, . . .. (i - 1)- i- cTi . , t
k kN

min Vk > 0,
t

Vk =
i=1

(cTi - Ri ),

59

Nt , . ,

(0, t) =
k=0

P(Nt = k )P(min Vi > 0).
ik

{mini k Vi > 0} k=1 {cTi - Ri > 0}, , i c = maxi l2 ci . ci P(ci Ti - Ri > 0) = 1 - > 0. ,

(0, t) >
k=0

P(Nt = k )(1 - )k = E(1 - )Nt .

, , Nt , .. .

(0, t) >
k=0

(1 - )

k

(t)k -t e = e- k!

t

.

, , t , , , .. . , . t . 2 . , , , c. ^ t Zt = -Zt = Yt - ct. l1 ^ lim t-1 Zt = · - c t l2 µ ^ 1. t-1 EZt . . 3. , . 1 Nt , t Zt = Xt - Yt , Xt = i=1 Ci , a 2 Nt Yt = i=1 Ri . Nt1 Nt2 l1 l2 , , Ci Ri i- . {Ci } {Ri } m1 m2 , Zt = lim lim t t t Xt Yt - t t m 1 1 m 2 2 = 1· - 2· . l µ1 l µ2

µk = E l1 Tik , k = 1, 2, Tik k - i=1 , 1 = m1 ECi 2 = m2 ERi . i=1 i=1

5

, , 60

.., ..

. (., , [12]), . . , , . ( ) . , , , [4], , [15]. , , - , , , [23] (. [17], [6]).

[1] .., .., , .: - , 1980. [2] .., , .: , 2009. [3] .., // , 2003, . 10, . 2, .276-286. [4] .., .., // , 2013, . 8, 3, . 1930. [5] .., .., .. (k1 , k2 ) // .., 2012, . 2(42), . 1618. [6] .., ., ., , .: , 2007. [7] . . .: , 1984. [8] . , . 2. .: , 1984. [9] Afanaseva L., Bulinskaya E., Multi-supplier systems with seasonal demand. In: B.Vallespir and T.Alix (eds.) Advances in Production Management Systems: New Approaches. Proceedings of the IFIP WG 5.7, Springer, 2010, p. 267274. [10] Arrow K., Harris T., Marschak J., Optimal inventory policy // Econometrica, 1951, v. 11, p. 250252. [11] Arrow K., Karlin H., Scarf H. (eds.), Studies in the Mathematical Theory of Inventory and Production. Stanford, California: Stanford University Press, 1958. 61

[12] Asmussen S., Applied Probability and Queues. 2nd ed. Springer-Verlag, New York, Berlin, Heidelberg, 2003. [13] Bel lman R., Dynamic Programming. Princeton, New York: Princeton University Press, 1957. [14] Bel lman R., Glicksberg I. and Gross O., On the optimal inventory equation // Management Science, 1955, v. 2, p. 83104. [15] Bulinskaya E., Optimal and asymptotically optimal control for some inventory models. In: A.N.Shiryaev et al. (eds.) Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics and Statistics, 33, chapter 8, Berlin: Springer-Verlag, 2012. [16] Cox D.R., Renewal Theory, Methuen and Company, Ltd. 1962. [17] Embrechts P., Kluppelberg C., Mikosch T., Modelling Extremal Events for Insurance and ¨ Finance, Springer-Verlag, Berlin, 1997. [18] Fel ler W., On the integral equation of renewal theory // Annals of Mathematical Statistics, 1941, v. 12, p. 722727. [19] Fel ler W., Fluctuation theory of recurrent events // Transactions of American Mathematical Society, 1949, v. 67, p. 98119. [20] Mitov K.V., Omey E., Renewal Processes, Springer, 2014. [21] Smith W. L., Renewal theory and its ramifications // Journal of Royal Statistical Society, Series B, 1958, v. 20, p. 243302. [22] Smith W. L., On the cumulants of renewal processes // Biometrika, 1959, v. 46, p. 129. [23] Zinchenko N. M., Strong limit theorems for the risk process with stochastic premiums // Markov Processes and Related Fields, 2014, v. 20, 3, p. 527544.

62

..

..
1

VaR, CVaR MINVaR .

1
1.1

. , , , . , , . 1974 . 2 . , . 1988 ( , I). 1996 VaR (Value at Risk; , ) . , VaR , ( ). J. P. Morgan 1980- .
, 011235813@inbox.ru, , , - . .. . 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , -. 19 .
1

63

1994 VaR. , RiskMetrics Group [10], [8]. VaR, , . , , [1] [2], : , ( ), . , , . VaR , , . CVaR (Conditional Value-at-Risk), [1, 2] TailVaR, ([14] - [17]). CVaR VaR , VaR. CVaR 2000 [13]. 2012 Fundamental review of the trading book consultative document , CVaR VaR. . .. MINVaR WVaR [4, 5, 6]. [4] WVaR CVaR. MINVaR . [20]. MINVaR [21]. , . . , MINVaR , , , .

1.2

. - X , (x(u) = FX 1 (u), = I [F ]):
- V aR (X ) = inf {x|FX (x) } = FX 1 (), C V aR (X ) = E[X |X > V aR (X )] = + 1 1

(1) 1 x(F ) dF = 1-
0

1 = 1-
V aR (X )

1 x dF (x) = 1- 64

x(F ) dF,

(2)

..

M I N V aRn (X ) = E[max{X1 , . . . , Xn }] = E[X
+ 1

(n:n)

]=
1

(3)
n-1

=
-

x dF n (x) =
0

x(F ) dF n =
0

x(F )nF

dF.

(1) , (2) . Y = -X : V aR (Y ) C V aR (Y )
- = -FY 1 (1 - ), = -E[Y |Y < -V aR (Y )] = -V aR (Y ) 1-

1 =- 1-
-

1 y dF (y ) = - 1-
0 (1:n)

y (F ) dF,

M I N V aRn (X ) = -E[min{Y1 , . . . , Yn }] = -E[Y
+ 1

]=

=
-

y d (1 - (1 - F (y ))n ) =
0

y (F )n(1 - F )n-1 dF.

. : X R, : 1. : > 0 (X ) = (X ); 2. : (X1 + X2 ) (X1 ) + (X2 ); 3. : X1 X
2

(X1 ) (X2 );

4. : a R (X + a) = (X ) + a. , C V aR (X ) M I N V aRn (X ) , [13] [4] . V aR (X ) [2].

1.3

: · , , · · V aR (X ), C V aR (X ) M I N V aRn (X ). . , (aX + b) = a(X ) + b a > 0, c EX = 0 DX = 1 ( ). (2) (3) , C V aR (X ) > 0 M I N V aRn (X ) > 0. 65

1. X , n-1 . 0 < M I N V aRn (X ) 2n - 1 [11]. 2. ( 2.1) X M I N V aRm (X ) = , n > m M I N V aRn (X ) n-1 (m - 1)(n + m - 1) M I N V aRn (X ) (2m - 1) + n-m 2n - 1 (2n - 1)(2m - 1) (m - 1)2 - 2m - 1
2

,

max ;

n-1 (m - 1)(n + m - 1)

(2m - 1) -

n-m 2n - 1

(2n - 1)(2m - 1)

(m - 1)2 - 2m - 1

2

.

3. ( 2.1) X M I N V aRm (X ) = , n < m M I N V aRn (X ) min ; n-1 (m - 1)(n + m - 1) M I N V aRn (X ) n-1 (m - 1)(n + m - 1) (2m - 1) + m-n 2n - 1 m-n 2n - 1 (2n - 1)(2m - 1) (m - 1)2 - 2m - 1
2

,

max 0;

(2m - 1) -

(2n - 1)(2m - 1)

(m - 1)2 - 2m - 1

2

.

4. ( 2.1) X M I N V aR2 (X ) = , M I N V aRn (X ) (n - 1) . 1. ( 2.1) X M I N V aR2 (X ) = , M I N V aRn (X ) min (n - 1) ; n-1 n+1 3 + n-2 2n - 1 3(2n - 1) 1 - 3
2

.

2. ( 2.1) X M I N V aRm (X ) = , n = 2 M I N V aR2 (X ) max 1 ;2 m-1 m -1 (2m - 1) - (m - 2) 2m - 1 3 (m - 1)2 - 2m - 1
2

.

5. ( 3.1) X , 0 < C V aR (X ) . 1- 66

..

[3]. 3.1 . 6. ( 4) X V aR (X ) C V aR (X ): 1- C V aR (X ) V aR (X ) C V aR (X ), V aR (X ), V aR (X ) > 0, C V aR (X ) - V aR (X ), V aR (X ) 0. 1- - C V aR (X ) [3]. 7. ( 3.2) X C V aR (X ) = , > C V aR (X ) + 1- - 1- 1- 1- , 1- - 2 , 1- 1- > 1- 1- . 1-

8. ( 3.2) X C V aR (X ) = , < 1- C V aR (X ) 1- 1- + 1- 1-

, - 1- 1- - 1-
2

>

,

1- . 1-

2-3 7-8, , . , (1 (X1 ), 2 (X1 )) 1 (X2 ), 2 (X2 ) 2 (X1 ) 2 (X2 ). ( ) 2 (·) 1 (·) (., , . 1-6). , , , , , , . 9. ( 5.3) X , [a, b], V aR (X ) min[V aR (X )] max[V aR (X )] 1 -1 - a(b - a) 0 a 1 + a2 a + (1 - )b 1 b2 1- - 2 2 1+a 1+b 1- b2 (1 - )b(b - a) - 1 < b 1 + b2 a + (1 - )b , a b . 67

2
2.1

M I N V aR

n

EX , DX M I N V aRm (X )
1

. x(F )nF
0 1 n-1

dF extr,

x(F )dF = 0,
0 1

x2 (F )dF = 1,
0 1 m-1

(4)

x(F )mF
0

dF = .

(5)

EX

(n:n)

, 0 = ±1:

L = 0 xnF n-1 + 1 x + 2 x2 + 3 xmF m-1 , 0 xnF n-1 + 1 x + 2 x2 + 3 xmF m-1 = 0, x 0 nF n-1 + 1 + 22 x + 3 mF m-1 = 0,
1

0 nF
0

n-1

+ 1 + 22 x + 3 mF

m-1

dF = 0,

0 + 1 + 3 = 0, 1 = -0 - 3 , 0 (nF n-1 - 1) + 22 x + 3 (mF m-1 - 1) = 0, 22 x = 0 (1 - nF n-1 ) + 3 (1 - mF m-1 ). a = 3 0 , b= : 22 22 x = a(1 - nF
n-1

) + b(1 - mF

m-1

).

(6)

:
1

x(F )nF
0

n-1

dF = -a

(n - 1)2 (n - 1)(m - 1) -b . 2n - 1 n+m-1

(7)

(6) (4) (5): a (n - 1)2 (n - 1)(m - 1) (m - 1)2 + 2ab + b2 = 1, 2n - 1 n+m-1 2m - 1 (n - 1)(m - 1) (m - 1)2 -b = . -a n+m-1 2m - 1
2

68

..

b: b
2

(n + m - 1)2 (m - 1)2 (n - m)2 (n - m)2 + 2b + 2 - 1 = 0. 2m - 1 (2n - 1)(2m - 1) (2n - 1)(2m - 1) (2n - 1)(m - 1)2

: (n - m)2 D = 4 (2n - 1)(2m - 1) a b: b= 2m - 1 (m - 1)2 - ± (2n - 1)(2m - 1) (n - m)2 (2n - 1)(2m - 1) (n - m)2 (m - 1)2 - 2m - 1
2

(m - 1)2 - 2m - 1

2

.

,

a=

n+m-1 (n - 1)(m - 1)

(m - 1)2 - 2m - 1

2

.

a b (7). :
1 n-1

x(F )nF
0

dF = n-m 2n - 1 (m - 1)2 - 2m - 1
2

=

n-1 (m - 1)(n + m - 1)

(2m - 1) ±

(2n - 1)(2m - 1)

.

, , ( ), -: f g dt EX
(n:n) 1

f dt

2

g dt

2

1 2

.

:
1 1 n-1 m-1

EX

(n:n)

=
0 1

x(F )nF

dF - c
0

x(F )dF -
0 m-1

x(F )mF

dF + = (8)

=
0

x(F ) nF

n-1

- c - mF

dF + .

EX
(n:n)

1

1

nF
n-1

1 2

-
0

x2 dF
0 2

- c - mF

m-1 2

dF =

=

n m2 nm + c2 + 2 - 2c - 2 + 2c . 2n - 1 2m - 1 n+m-1 69

EX

n2 m2 nm + c2 + 2 - 2c - 2 + 2c . 2n - 1 2m - 1 n+m-1 c , :
(n:n)

: ±

c = 1 - , = (n - 1)(2m - 1) 1 + (n + m - 1)(m - 1) |n - m| (2n - 1)(2m - 1)
(m-1)2 2m-1

.

-

2

c , , . c . , . 1: n > m :

EX

(n:n)

(2m - 1) + n-m 2n - 1 (2n - 1)(2m - 1) (m - 1)2 - 2m - 1
2

n-1 (m - 1)(n + m - 1) EX(n:n) max ;

.

n-1 (m - 1)(n + m - 1)

(2m - 1) -

n-m 2n - 1

(2n - 1)(2m - 1)

(m - 1)2 - 2m - 1

2

.

2: n < m :

EX

(n:n)

n-1 (m - 1)(n + m - 1) n-1 (m - 1)(n + m - 1) (2m - 1) + m-n 2n - 1 m-n 2n - 1 (2n - 1)(2m - 1) (m - 1)2 - 2m - 1
2

min ; EX
(n:n)

.

(2m - 1) - (2n - 1)(2m - 1) (m - 1)2 - 2m - 1
2

max 0;

.

n > m, m = 2 EX EX(2:2) : 70

(n:n)

,

..

= EX
n

(2:2)

1 1 = E[max(X1 , X2 )] = E[X1 + X2 + X1 - X2 ] = E X1 - X2 , 2 2
n

Xi +
i=1 i
Xi - X

j

=
i=1

X
n-1

(i:n)

+
i
X

(i:n)

-X

(j :n)

n-1

i=1

X

(i:n)

+X

(n:n)

+
i=1

(X

(n:n)

-X

(i:n)

) = nX

(n:n)

,

X

(n:n)

1 n

n

Xi +
i=1

1 n

|Xi - Xj |,
i
EX

(n:n)

1 n

E X i - Xj =
i
n-1 12 C n E X1 - X2 = · 2 = (n - 1) . n 2

, , EX
(n:n)

min (n - 1) ;

n-1 n+1

3 +

n-2 2n - 1

3(2n - 1)

1 - 3

2

.

n = 2, m > n, : EX
(2:2)

max

1 ;2 m-1 m -1

(2m - 1) - (m - 2)

2m - 1 3

(m - 1)2 - 2m - 1

2

.

EX(3:3) = EX(2:2) : , 2 , ( ).

. 1: , 2 , , n = 3. 1 n-1 2(n + 1) (n + 1)(3n - 1) . 71

n = 2, m = 3 . 1.

2.2

EX , DX C V aR (X )

1

EX

(n:n)

=
0

x(F )nF

n-1

dF

1 1 1

EX =
0

x(F ) dF = 0,

DX =
0

x2 (F ) dF = 1,

1 C V aR (X ) = 1-
0

x(F ) dF = ,

-:
1 1 1

EX

(n:n)

- =
0

x(F )nF
1

n-1

dF - c
0
1 2

x(F ) dF - 1-
0

x(F ) dF,
2

1

1 2

EX

(n)

-

0

x2 (F ) dF
0

nF

n-1

-c-

1 1-

dF =

=

(n - 1)2 1 1 - n + c2 + 2 - 2c - 2 + 2c . 2n - 1 1- 1- 1 - n + 2c . 1-

1 (n - 1)2 + c2 + 2 - 2c - 2n - 1 1- ( ) c , : EX
(n:n)

: ±

2

c = 1 - , = (1 -
n-1

)

1-

(n-1)2 2n-1

-

(1-n-1 ) 1- 2

2

1-

-

.

c 1- - 1- 72 (n - 1)2 (1 - n-1 ) - 2n - 1 1-
2

1-

n-1

±

2

.

..

. 2:

. 3:

. 4:

. 5:

3
3.1

C V aR

EX DX

( ), - C V aR (X ):
1 1 1

1 C V aR (X ) = 1-
0

x(F ) dF - c
0

x(F ) dF =
0 2

x(F )
1 2

1 -c 1-

dF,

C V aR (X )
0

1

1 2 x2 (F ) dF
0

1

1 -c 1-

dF =

1 - 2c + c2 . (9) 1-

(9) c, : c = 1, C V aR (X ) 73 . 1-

C V aR (X ) > 0 0 < C V aR (X ) . 1-

(): Z = {z ; -1/z } (. 5.2); z - p(z ) 0 C V aR (Z ) 0 . C V aR , , > 0.5 : - p = 1 - , 1- X= 1- 1 - p = .

3.2

EX , DX C V aR (X )
1

1 = I [F ] C V aR (X ) = 1-
0

x(F ) dF = . -

- C V aR (X ):
1 1 1

1 C V aR (X ) - = 1-
0 1

x(F ) dF - c
0

x(F ) dF - 1-
0

x(F ) dF,

=
0

x( F )
1

1 1 -c- 1- 1-
1 2

dF
2

1

1 2

C V aR (X ) -

0

x2 (F ) dF
0

1 1 -c- 1- 1-

dF .

1: > C V aR (X ) - 1 1 2 + c2 + 2 - 2c - + 2c . 1- 1- 1-

( ) c , : c = 1 - , =1±
1- 1- 1-

-

1- 2

-

. 1- 74 - 1- 1- - 1-

extr{C V aR (X )} = ±

2

.

..

2: < C V aR (X ) - 1 1 2 + c2 + 2 - 2c - + 2c . 1- 1- 1-

( ) c , : c = 1 - , 1- ± = 1 -
1- 1- 1-

-
1- 2

-

. 1- - 1- 1- - 1-

extr{C V aR (X )} =

1-

± 1-

2

.

: < C V aR (X ) C V aR (X ) ( 1- ) C V aR (X ) C V aR (X ) ( (10) q q , 1- E [X |X q ] E [X |X q ]). > .

. 6:

4

V aR (X ) C V aR (X ) EX = 0 DX = 1
EX = E[X |X q ] + (1 - )E[X |X > q ],

10.

q - - X . 75

.
+ q +

EX =
-

x dF (x) =
- q

x dF (x) +
q

x dF (x) =
+

1 =·
-

1 x dF (x) + (1 - ) · (1 - )
q

x dF (x) =

1 1 E[X · I (X q )] + (1 - ) E[X · I (X > q )] = = P (X q ) P (X > q ) = E[X |X q ] + (1 - )E[X |X > q ].

3. E[X |X q ] = 1 (EX - (1 - )E[X |X > q ]) . (10)

11. E[X |X q ] 1 (EX - (1 - )E[X |X > q ]) q q E[X |X > q ], E[X |X > q ]. (11)

4. X : 1- C V aR (X ) V aR (X ) C V aR (X ), V aR (X ), V aR (X ) > 0, C V aR (X ) - V aR (X ), V aR (X ) 0. 1- - (12) (13)

. (12) (11) q V aR (X ) E[X |X > q ] C V aR (X ). (12) C V aR : C V aR (X ) V aR (X ), V aR (X ). C V aR (X ) - 1-

76

..

. 7:

5

V aR

FX (·) - V aR (X ) = FX 1 () FX (z ) z . FX (z ) = E[I (X z )].

5.1

FX (z )
b

a

I [t z ] d (t) n + 1 (t) d (t),
T

, ( V (c0 ), c0 (n + 1)- , c0 = i
T

ui (t) d (t)).

[18] ( XII, §2 2.1) , Imax = sup
V (c0 ) T

(t) d (t)

77

n

U (t) =
i=0 n

a0 ui (t) i

P+ =

u(t) =
i=0

ai ui (t) u(t) (t)
n

, (t) , Imax = inf
i=0

ai c 0 . i

2.3 ([18], XII, §2), , , Imax =
T n

d ,

S= , {ui }
n i=0

tT
i=0

a0 ui (t) = (t) . i

T -, n+1 ui n , S +1 2 . P- , (t) .
b

a

I [t z ] d (t)

ui = ti , i = 0, . . . , n. P+ n. . n = 2, P+ [a, b] I [t z ], S 2, 3 . : 1. Z . 2. 2, I [t z ] , , Z . 3. .2
n

inf
i=0

ai c0 = E[U (Z )] = Imax . i

. 78

..

5.2

- Z EZ = 0 DZ = 1. 2 [9]. z = -1/z . [a, b]: z1 = z [a, b], p1 = 1 , 1 + z2 z2 z2 = z [a, b], p2 = . 1 + z2

Z = {z1 , z2 , z3 } : 1 + z2 z3 (z2 - z1 )(z3 - z1 -(1 + z1 z3 ) p2 = (z2 - z1 )(z3 - z2 1 + z1 z2 p3 = (z3 - z1 )(z3 - z2 p1 = z1 < z3 < z2 < z1 < z3 . , , .

) ) )

5.3

FX (z ) V aR (X ) EX DX

Fmin (z ) Fmax (z ) F (z ) = E[I (X z )], , z Fmin (z ) F (z ) Fmax (z ), . 5.1 Fmin (z ) Fmax (z ) 1 azb 0 {z , z } 1 + z2 1 + bz 1 + az b

(1): {z , z }, z (a, b). U (x) : U (z ) = 1, U (z ) = 0, U (z ) = 0. (x - z )2 U (x) = . (z - z )2 L(x) 0. (2): {a, z , b}, z (b, a). U (x) : U (a) = 1, U (z ) = 1, U (b) = 0. (b - x)(x + b - z - a) . U (x) = (b - z )(b - a) L(x) : L(a) = 1, L(z ) = 0, L(b) = 0. (b - x)(x - z ) . L(x) = (b - a)(z - a) (3): {z , z L(x) }, z (a, b). U (x) 1. :

L(z ) = 1, L (z ) = 0, L(z ) = 0.

(x - z )2 . L(x) = 1 - (z - z )2 L(x) 0. : x = b, L(x) 1, U (x) 1. 5.1 E[U (Z )] E[L(Z )] . Fmin (x), Fmax (x) V aR (X ) : 80

..

. 8: Fmax (x) = , x = min[V aR (X )], Fmin (x) = , min[V aR (X )]. . 5.2 min[V aR (X )] max[V aR (X )] 1 -1 - a(b - a) 0 a 2 1+a a + (1 - )b 2 b 1 1- - 2 2 1+a 1+b 1- b2 (1 - )b(b - a) - 1 < b 1 + b2 a + (1 - )b .. , , .. .

[1] Artzner P., Delbaen F., Eber J.-M., Heath D., Thinking coherently // Risk, 1997, v. 10, p. 6871. [2] Artzner P., Delbaen F., Eber J.-M., Heath D., Coherent Measures of Risk // Mathematical Finance, 1999, v. 9, p. 203228. [3] Bertsimas D., Lauprete G.J., Samarov A., Shortfall as a risk measure: properties, optimization and applications // Journal of Economic Dynamics and Control, 2004, v. 28, 7, p. 1353-1381. [4] Cherny A., Weighted V@R and its properties // Finance and Stochastics, 2006, v. 10, p. 367393. [5] Cherny A., Madan D., On Measuring the Degree of Market Efficiency. Preprint, 2007. [6] Cherny A., Orlov D., On two approaches to coherent risk contribution // Mathematical Finance, 2010, v. 21, 3, p. 557 571. 81

[7] David H.A., Nagaraja H.N., Order statistics (3rd edition). Wiley, 2003. [8] Guldimann T.M., The story of RiskMetrics // Risk, 2000, v. 13, 1, p. 56-58. [9] Jansen K., Haezendonck J., Goovaerts M.J., Analytical upper bounds on stoploss premiums in case of known moments up to the fourth order // Insurance: Mathematics and Economics, 1986, v. 5, p. 315334. [10] RiskMetrics Technical Document (4th edition). New York, J.P.Morgan, 1996. [11] Hartley H.O., David H. A., Universal bounds for mean range and extreme observation // The Annals of Mathematical Statistics, 1954, v. 25, p. 8599. [12] Kaas S., Goovaerts M.J., Extremal values of stop-loss premiums under moment constraints // Insurance: Mathematics and Economics, 1986, v. 5, p. 279283. [13] Pflug G., Some Remarks on Value-at-Risk and Conditional-Value-at-Risk / Probabilistic Constrained Optimisation: Methodology and Applications. Dordrecht, Boston: Kluwer Academic Publishers, 2000. [14] Rockafel lar R.T., Uryasev S., Optimization of Conditional Value-At-Risk // The Journal of Risk, 2000, v. 2, p. 2142. [15] Rockafel lar R.T., Uryasev S., Conditional value-at-risk for general loss distributions // Journal of banking & finance, 2002, v. 26, 7, p. 14431471. [16] Uryasev S., Conditional Value-at-Risk: Optimization Algorithms and Applications // Financial Engineering News, 2000, v. 14, p.15. [17] Uryasev S., Rockafel lar R.T., Conditional value-at-risk: optimization approach // Stochastic optimization: algorithms and applications, 2001, p. 411435. [18] ., ., . .: , 1976. [19] .., .., . .: , 1973. [20] .., // , 2008, . 53, 1, . 168172. [21] .., , , // , 2013, . 8, 3, . 124136.

82

..

..
1

. , , , . , , , . .

1

[6] [7] . . HARA-, , -U (x)/U (x) = 1/(ax + b), U (x) , a b . . , U (x) , . , -. [14], , , . , , , , . , , , , . , , (
, m_y_ivanov@mail.ru, , , - . .. .
1

83

supxR U (x), , ), [1, 2, 10, 18, 19]. . [12]. , , [3, 5]. , . , , . [12] . , . . , , . , , [12], , , , , , [6, 7]. , [3, 5]. , [6, 7]. (, F , F, P ), F = (Ft )0tT , S , [0, T ]. S . (x, H ), x , H , S . H . X = (Xt )0tT
t

Xt = x +
0

Hu dSu , 0 t T .

, , : S = E (L), L > -1, L 0. dS = S- dL, , ,
t

Xt = x +
0

Hs dLs , 0 t T .

- X (x) = {X : X = x + H · L, H L}. (1)

x T : u(x) = sup E[1 - exp(-XT )].
X X (x)

(2)

84

..

(B , C, L ), Bt ( ) = bt, Ct ( ) = ct, L ( , dt, dx) = dt (dx), (3)

b, c (c 0) , , (1 x2 )d < , - : Ee
iuLt

1 = exp t ibu - cu2 + 2

(eiux - 1 - iuh(x))d

,

h(x) = x1|x|1 . (b, c, ). L , c = 0, [x < 0] = 0, b - x1|x|1 (dx) 0, c = 0, [x > 0] = 0, b - x1|x|1 (dx) 0, . [17]. L , , . [6].

2

, X . : X (x) = {X : X (1), H - X }.

X , exp(X - X0 ) , . X (x) = X (1) + x - 1, X (1). 1. X (1) (2) L , L X = 1 + y L, y = min{y : -b + cy + (h(x) - xe
-y x

) (dx) 0}.

F (b, c, ) L: F (y ) = -b + cy + (h(x) - xe-yx ) (dx).

1. F : 1. F . y0 = sup{y : x>1 xe-yx (dx) = } (sup = -), y0 [-, 0]. F (y ) y > y0 , y < y0 F (y ) = -. y0 F , , -. 85

2. F , y , 0 < F (y ) < + y = y0 . y . 3. , F , y < y0 xe-yx (dx) x>1 e-yx (dx), y y0 . x>1 . -b + cy R, c > 0. I1 (y ) - I2 (y ), I1 (y ) =
|x|1

x 1 - e- xe
x>1 -y x

yx

(dx),

I2 (y ) =

(dx).

, , {0} = 0, (x2 1) (dx) < . (4)

I1 . , , K (y )x2 , K (y ) > 0 . I1 y . , y1 y2 x(1 - e-y1 x ) x(1 - e-y2 x ). I1 y R. I2 , y . y > 0, . (4) . y 0. , , . y0 = sup{y : I2 (y ) = +}. , , y < y0 I2 , y > y0 , , F (y0 ) , F (y0 ) F . y0 = -, I2 R. I2 y , y0 = -, y > y0 , y0 . , F , , y y > y0 - y < y0 . y0 F , F . , F y > y0 , , L , . . 2. , y - +. y = -. , F (y ) 0, y R. , L , c = 0 -b + h(x) (dx) 0. L. y = +, F (y ) < 0, y R. L , c = 0 -b + h(x) (dx) 0. , L . , y . y0 > - F (y0 ) = -, limyy0 + F (y ) = -, . F (y0 ) > 0 limyy0 + F (y ) = F (y0 ) > 0. , F y y0 , F y0 y . 3 , y < y0 xe
x>1 -y x

(dx)

86

..

e-yx (dx)
x>1

y < y0 . F , F y0 y y0 . . D y , x>1 e-yx (dx) < C [6, 7]. 1, , y > y0 . D [y0 , +), x>1 e-y0 x (dx) < , (y0 , +), x>1 e-y0 x - 1 [6]. . y < y0 (-, +), y0 = -, (dx) = .

2. X , Ht D dP dt .., 0 t T . . 2.13 [13], , -X , , J
-X

= 1x>1 ex

-X

, -X -X . [11], -H · L G B (R \ {0})
s

-X

([0, s] â G) =
0

dt

1G (-Ht x) (dx), 0 s T .

s

J

-X s

=
0 s

dt dt
0

1-

Ht x>1

exp(-Ht x) (dx) (5)

=

exp(-Ht x) (dx) , 0 s T .
-Ht x>1

Ht D (5) D. , / -X J , , .. , t, Ht ( ) D, 0. , / Ht D dP dt .. (. [15]), , . 3. U (x) = 1 - e : 1. e-yx d < .
x>1 -x

X = 1 + y L, y R

(6)

2. EU (Xt ) > -, 0 t T . 3. -X . 87

. , 2 1. , (6) . y < 0 R. Gt = t, µL L. exp(-y L) 0 t T : 1 exp(-y L) = E (-y L + y 2 c · G + [e-yx - 1 + y x] (µL )) 2 1 = E (-y b · G - y · B + y 2 c · G - y h (µL - L ) 2 -y x + [e - 1 + y h]1|x|1 (µL ) + [e-yx - 1]1x>1 (µL )). = E (D + A) = E (D)E (A), At = [e-yx - 1]1x>1 (µL )t , Dt . , A D . E exp(-XT ) < , EE (A)T < . EE (A)T = E(1 + E (A- ) · AT ) E(1 + AT ), EAT < . , [e-yx - 1]1x>1 (µL ) [e-yx - 1]1x>1 (L ). , [e-yx - 1]d , x>1 e-yx d . x>1 , 1 2, (6). x>1 (e-yx - 1)d . . , A . x2 0, [e-yx - 1 - y h]1|x|1 (µL ) . , : exp(-1 - y L) = E (-1 - y · B + (e
-y x

- 1) (µL - L ) + (y ) · G) = Z exp(-1 + (y ) · G), (7)

1 (y ) = -y b + 2 y 2 c + [e-yx - 1 + y h] . , 8.30 [8] Zt = E (-y · Bt + (e-yx - 1) (µL - L )t ) , , . = -y , Y (x) = e-yx . E exp(-y L)T = exp(T (y )) < , . , 3 1. (5) 2, Ht = y , y R. y D, / t. y D, t D1 < +, 0 t D1 t . , y D X . D 1 (6), .

3 Z , Q. C III.3.24 [9] L (B , C , L ) . Bt = b t, Ct = c t, L ( , dt, dx) = dt (dx), b = b - cy + h(x)(e- c = c, = Y = e-yx , 88
yx

- 1) , (8)

..

. L Q. . , U (x) = 1 - e-x XT , XT :
U (XT ) U (XT ) + (XT - XT )U (XT ).

(9)

, X X , X X E(XT - XT )U (XT ) 0 EU (XT ) > -, X P . U E(XT - XT ) exp(-XT ) 0.

(10) y D, E(1 - e-Xt ) > -, 0 Q.

, , X = 1 + y L . 3 , , t T exp(-XT ) = ZT exp( (y )) = exp( (y ))ZT , ZT

E(XT - XT )U (XT ) = exp( (y ))EQ (XT - XT ).

, xd =
x>1 x>1

xe

-y x

d

F y (. 1), EQ XT < (8) EQ XT = 1 + T y (b + x>1

xd ) = 1 + T y F (y ).

EQ XT . X X , 2 Ht D dP dt .. X = H · L . H X c + h (µ - ) , (. I.4.2, I.4.40 [9]), H · (X c + h (µ - )) . EQ H · (X c + h (µ - )) = 0. , EQ XT . Gt = t, EQ XT = EQ (H · (b G + (x - h) µ))T . (x - h) , T EQ (H · (b G + (x - h) ))T = 0 Lt dt, Lt = Ht (b - cy + =Ht (b - cy + (xe h(x)(e
-y x -y x

- 1) (dx)) +

Ht (x - h)e-

y x

(dx)

- h(x)) (dx)) = -Ht F (y ).
T

H , EQ 0 Lt dt, , EQ XT . T EQ (XT - XT ) = -F (y )E( 0 (Ht - y )dt). F (y ) = 0, EQ (XT - XT ) = 0. F (y ) = 0, 2 1 |y | < 0 < F (y ) < , y D. Ht y .., EQ (XT - XT ) 0. , X (10) . (9) . . 89

, X (x), . . , X (x) , n X n X (x), n + X n + .., EU (XT ) .. U (+), u(x) = 1. , [12, 19], Xb (x) x + H · L, . [12] (2) , y 0 : F (y ) = 0, X n , X n x + y L, -n. 1, F , L Q. y 0 , L . , Xb (x). X (x), L, F . X , , X (x), . F . , 1 H D T , F , , 0 (Ht - y )dt . , 2, H D , X . F X 1 X (x) = {X : X (1), H }. , X = E (y L), y , 1 (. [6, 7]). . y . , , , , .

3

(2) : v (y ) = inf E V (YT ),
Y Y (y )

(11)

V (y ) = supx>0 (U (x) - xy ), y > 0. U (x) = 1 - exp(-x) V (y ) = 1 - y + y ln y . [19] Y (y ) M(y ) Y , YT /y 90

..

P . , F . v (y ) = =
Y M(y ) Y M(y )

inf

E (YT ln YT - YT + 1) E (YT ln YT ) - y + 1

(12) (13)

inf

= y ln y + y v (1) - y + 1. , M(y ) = y M(1) EYT = y . , (12) y = 1 y. (13) y = 1 , . , , [3], Q, L . , L . , F y = 1 C0 (t) exp(-y Lt ), C0 (t) = exp(- (y )t) , t ( , (7)). . , [5], , , [16], , . V (y ) = y ln y YT , YT V (y1 ) - V (y2 ) V (y2 )(y1 - y2 ), YT (ln YT - 1) YT (ln YT - 1) + (YT - YT ) ln YT . , , , Y . , Y = C0 (t) exp(-y L) . , EYT (ln YT - 1) < , Y M(1) E(YT - YT ) ln YT 0. (7) , E(1 + y LT ) exp(-1 - y LT ) = E exp(-1 + (y )T )EQ XT = E exp(-1 + (y )T ), Q Zt = E (-y · Bt + (e-y x - 1) (µL - L )t ). , EYT ln YT < . , , , F , Q EQ LT = 0, L . , Y M(1). , E(YT - YT ) ln YT 0. Y Y , E(YT - YT ) ln YT = E(YT - YT )(C0 (T ) + 1 - XT ) = 0, . , Yt = C0 (t) exp(-y Lt ) . , y0
YT = y0 U (XT ),

V (y ) = 1 - y + y ln y v (y ) (11) u(x) v (y ) = sup[u(x) - xy ].
x 0

91

. , F . . [4], , , (13) . , (13). , F , [4] Y . .. .

[1] Biagini S., Sirbu M., A note on admissibility when the credit line is infinite // Stochastics An International Journal of Probability and Stochastic Processes, 2012, v. 84, 23, p. 157169. [2] Delbaen F., Grandits P., Rheinl¨ ander T., Samperi D., Schweizer M., Stricker C., Exponential hedging and entropic penalties // Mathematical finance, 2002, v. 12, 12, p. 99123. [3] Essche F., Schweizer M., Minimal entropy preserves the L´ property: how and why // evy Stoch. Proc. Appl., 2005, v. 115, 2, p. 299327. [4] Gushchin A. A., Khasanov R. V., Morozov I. S., Some functional analytic tools for utility maximization // Modern Stochastics and Applications, 2014, v. 90, p. 267285. [5] Hubalek F., Sgarra C., Esscher transforms and the minimal entropy martingale measure for exponential L´ models // Quantitative Finance, 2006, v. 6, 2, p. 125145. evy [6] . ., // , . 1. . , 2014, 6, . 1624. [7] . ., // , 2014, 4, . 781790. [8] Jacod J., Calcul stochastique et probl´mes de martingales // Berlin, Heidelberg, New e York: Springer, 1979. [9] Jacod J., Shiryaev A. N., Limit theorems for stochastic processes, 2nd edition, Springer, 2003. [10] Kabanov Y. M., Stricker C., On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper // Mathematical Finance, 2002, v. 12, 2, p. 125134. [11] Kal lsen J., A didactic note on affine stochastic volatility models, From stochastic calculus to mathematical finance // Springer, Berlin, 2006, p. 343368. [12] Kal lsen J., Optimal portfolios for exponential L´ processes // Mathematical Methods evy of Operations Research, 2000, v. 51, p. 357374. 92

..

[13] Kal lsen J., Shiyaev A. N., The cumulant process and Esscher's change of measure // Finance and Stochastics, 2002, v. 6, 4, p. 397428. [14] Kramkov D., Schachermayer W., The condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets // Annals of Applied Probability, 1999, v. 9, 3, p. 904950. [15] . ., // , 1970, . 15, 2, . 330336. [16] Miyahara Y., Option pricing in incomplete markets: Modeling based on geometric L´ evy processes and minimal entropy martingale measures. Imperial College Press, 2012. [17] Sato K.-I., L´ evy processes and infinitely divisible distributions. UK, Cambridge, Cambridge University Press, 1999. [18] Schachermayer W., A super-martingale property of the optimal portfolio process // Finance and Stochastics, 2003, v. 7, 4, p. 433456. [19] Schachermayer W., Optimal investment in incomplete markets when wealth may become negative // Annals of Applied Probability, 2001, v. 11, 3, p. 694734.

93

..1 , ..2 , ..3

, .. , , .. . , . ( ) - . , .

1

, . , [1]. cn , 0 2 p = 1/2, an = c1 c2 . . . ci . . . cn . an , n . , an , , 1, .. ci , an , . , n an 2n . 2-n . , , an < an >= 0 â (1 - (1/2)n ) + 2n â (1/2)n = 1 , < a2 >= 0 â (1 - (1/2)n ) + 2 n
1

(1)

2n

â (1/2)n = 2n .

(2)

, illarionov.ea@gmail.com, , , , - . .. . 2 , sokoloff.dd@gmail.com, , , - . .. . 3 , vntutubalin@yandex.ru, , , - . .. .

94

.., .., ..

, , < a2 >. n , [2]. , (x). () U (x). , U 2 . = U . t (3)

U , = exp(U t) ( (0, x) = 1). , < >= exp( 2 t2 /2), .. . U , , ,
n

(n , x) = exp
i=1

Ui (x) exp( n 2 ) exp( t ) .

(4)

Ui U , , . , . , , , k < k >, k . . , (3). , [2], [3], . . H(t, x) v(t, x) . , dH ^ = HA , dt (5)

^ A vi / xj ( ). v , 2 , 3 , . . . ^^ ^ H(n ) = H(0)B1 B2 . . . Bn , (6)

^ Bi , ^ . , A ^ ^ ^ ^ A(t) = Ai i- , Bi = exp(Ai ). , ^ Bi [4]. ^ ^ Bi , .. det Bi = 1. ^ , Bi i. 95

, (6) . , , . [5], [6]. [7]. , , (4), . , n e1 , e2 , e3 , H e1 n ( ). < |H|p >. , p. [5] , , . . , , . [8], [9].

2

, . , , (5). , , H . , Vij , - Vij = vi (x), vj (y) = r ri rj v2 (F (r2 )ij + F (r2 )(ij - 2 )) , 3 2 r (7)

r = y-x . < vi (x), vi (y) >= v 2 f (r2 ) f (r2 ) = exp(-r2 ). F (r2 ) = exp(-3r2 /5l2 ), .. l , l/v . v ^ , An . ^ , An . ^ , n. An 96

.., .., ..

3 â 3, B 9 â 9, , corr(Aik , Aj p ) = B3(i-1)+k,3(j -1)+p . Vij xk yp r = y - x 0 ( v 2 /5l2 ) 2 0 0 0 -1 0 0 0 -1 0 4 0 -1 0 0 0 0 0 0 0 4 0 0 0 -1 0 0 0 -1 0 4 0 0 0 0 0 0 0 2 0 0 0 -1 . B = -1 0 (8) 0 0 0 0 0 4 0 -1 0 0 0 -1 0 0 0 4 0 0 0 0 0 0 0 -1 0 4 0 -1 0 0 0 -1 0 0 0 2 ^ , A ^ ^^ ^ QAQT , Q . , , ^ exp A, . , . 1 ^ E log w exp A , (9) 2 w = (x, y , z ) , ( ). (9) , . ^ , A . , (9) . 2 , = = 1^ 1 ^^ E log w(I + A + A2 2 ) , 2 2 (10)

^ I . ^ , A p: 1 ^ log(E w exp A 2p , : p = p =
p

),

(11)

1 1^ ^^ log(E w(I + A + A2 2 ) p ) . 2p 2 97

(12)

, (11) , . p . ^^ ^ w(I + A + A2 2 /2) 2 = 1 + + 2 , 2 , ^ A, , 1 x2 + y 2 + z 2 = 1. :

= 2(A11 x2 + A12 xy + A21 xy + A13 xz + A31 xz + +A22 y 2 + A23 y z + A32 y z + A33 z 2 )

(13)

. (8) E2 = 8v 2 /5l2 ( E = 0) E = 2v 2 /l2 . , = 1 1 3v 1 E log(1 + + 2 ) = E( + 2 - 2 2 ) = . 4 4 2 10 l (14)

p = 1 1 (3 + p) v log E(1 + + 2 )p/2 = log(1 + (3p + p2 ) 2 ) = . 2p 2p 10 l (15)

, p, . , (15) , (14), p = 0. : 2 = 1v , 2l (16)

, 3v /4l, [4]. , , (5). . ^ = wA , (it)p p (t) = 1 + E . (17) p! p=1 (15) , E p = 1 + (3 + p)p 2 /5 2 . (17)

(t) = 1 +
p=1

(it)p (1 + (p2 + 3p) 2 ) = eit (1 + 4i 2 t - 2 t2 ) , ~ ~ ~ p! 98

(18)

.., .., ..

= / 5. ~ , - . (t) , - : (x) = (1 - x) - 4 2 (1 - x) + 2 (1 - x) . ~ ~ (19)

= E log /2 , 1 1 E log = 2 2
+

log(x) (x) dx =
-

3 2 3 ~ = , 2 10

(20)

(14).

3

, . , .. 80- . , , , , , , . , , . , , , . .

[1] .., . ., . ., . ., // , 1987, . 30, 5, . 353369. [2] Zeldovich Ya. B., Ruzmaikin A. A., Molchanov S.A., Sokoloff D. D., Intermittency of passive field in random media // Sov. Phys. JETP, 1985, v. 62, 6, p. 11881194. [3] Sokoloff D., Rubashny A., Small-scale dynamo in Riemannian spaces of constant curvature // GAFD, 2013, v. 107, 4, p. 403410. [4] Zeldovich Ya. B., Ruzmaikin A. A., Molchanov S. A., Intermittency, diffusion and generation in a nonstationary random medium // Sov. Sci. Rev., 1988, Sec. C, v. 7, p. 1110, Harwood Acad. Publ. [5] Furstenberg H., Noncommuting random products // Trans. Amer. Math. Soc., 1963, v.108, p. 377428. [6] Zeldovich Ya. B., Ruzmaikin A. A., Molchanov S. A., Kinematic dynamo in the linear velocity field // J.Fluid Mech., 1984, v. 144, p. 111. 99

[7] Tutubalin V. N., A central limit theorem for products of random matrices and some of its applications // Symposia Mathematica., 1977, v. XXI, p. 101 116. [8] E. A., . ., . ., // , 2012, . 13, . 218225. [9] E. A., // , 2013, . 14, . 3843.

100

..

..1

. , . , . , , , . . , , , .

1

, [8] 1903 . . 1957 [5] , , .. . , , (., , [4], [7], [12]). , . , , [1], [2]. , 1 , . , , [15], [10], [9]. , 1,
, karanar@mail.ru, , , . .. .
1

101

0, 1 2, . , . [6] , , .. , . . , . [16] . , , , . , [11].

2

. , : n N , , n, . x n . 1, zi : P(zi = 0) = p, P(zi = 1) = r, P(zi = 2) = q , p + q + r = 1, .. 1 p, 1 q r. i- : Sx (i) = min(Sx (i - 1) + 1 - zi , n), Sx (0) = x. x = min(i : Sx (i) < 0) x n. x Ex r 1. . 1. Ex = Ex x n x p = q -x2 + (2n + 1)x + 2(n + 1) Ex = , (1) 2p p = q Ex = x+1 p + q - p (q - p)
2

q p 102

-n-1

-

q p

x-n

.

(2)

..

. Ex Ex = pE E-1 = 0, En = E
n+1 x+1

+ q Ex

-1

+ rEx + 1 0 x n .

(3)

(3) (p = q p = q ), . p = q (3) Ex = 2 - xp , A + B x. 2 A B . A-B = 1 , 2p 2 2 - n = B - (n + 1) . 2p 2p
x p = q q-p , A + Bp = 1 q q- n B q =B p

A + B

q p

x

.

, p q p

n+1

+

1 . q-p

A B , (1), (2) Ex . x (., , [13]). ux,k = P(x = k ) , k x n x. ux,k = pu
x+1,k-1

+ q ux

-1,k-1

+ ru

x,k-1

0 x n k 1

(4)

u
n,k

=u

n+1,k

, u-

1,k

= 0 k 1; u-

1,0

= 1, ux,0 = 0 0 x n. ux,k sk n-x (s) 2 n+1 (s) 2

2. Ux (s) = Ux (s) = 1,2 (s) q p
x+1

k=0

n (1 (s) - 1)1 -x (s) + (1 - 2 (s)) (1 (s) - 1)n+1 (s) + (1 - 2 (s)) 1

,

(5)

ps2 (s) + (rs - 1)(s) + q s = 0.

. (4) sk k , Ux (s) = psU
x+1

(s) + q sUx-1 (s) + rsUx (s).

(6)

U-1 (s) = 1, Un (s) = Un+1 (s). 103

Ux (s) x (s), (6) (s) = ps2 (s) + q s + rs(s) 1,2 (s) = 1 - rs ± (1 - rs)2 - 4pq s2 . 2ps

s 1,2 (s). Ux (s) = A(s)x + B (s)x , A(s) B (s) 2 1 A(s) + B (s) = 1, 1 2 n A(s)1 + B (s)n = A(s)n+1 + B (s)n+1 . 2 1 2 , A(s) = 1 n+1 (1 - 2 ) 2 n+1 (1 - 1)1 + (1 - 2 )
n+1 2

B (s) =

2 n+1 (1 - 1) 1 n+1 (1 - 1)1 + (1 - 2 )

n+1 2

.

q , 1 2 = p ,

Ux (s) =

x+1 n+1 1 2

(1 - 2 ) + n+1 x+1 (1 - 1) 1 2 = n+1 (1 - 1)1 + (1 - 2 )n+1 2

q p

x+1

(1 - 1)n-x + (1 - 2 ) 1 (1 - 1)n+1 + (1 - 2 ) 1

n-x 2 n+1 2

,

(5) . 3. Q= (µ1 - 1)µm + (1 - µ2 )µm 2 1 , µ µ1 - µ2
1,2

=z±

z 2 - 1,

Pm (z ) c zj,m = cos . , µm - µm = (z + 1 2
m

2j +1 2m+1

, j = 0, . . . , m - 1.

z 2 - 1)m - (z -
m

z 2 - 1)m =
k

=
k=0

m z k

m-k

(z - 1) -
k=0

2

k 2

m z k

m-k

(z 2 - 1) 2 (-1)k =

= 2 z2 - 1 Q [m] 2 Q=
j =0

[

m-1 2

]

j =0

m z 2j + 1

m-2j -1

(z 2 - 1)j ,

m+1 z 2j + 1

[
m-2j

m-1 2

]

(z - 1) -
j =0

2

j

m z 2j + 1

m-2j -1

(z 2 - 1)j = Pm (z ).

(7)

m, z . Pm (z ). 104

..

z -1. µ2 -1 µ1 < 0. µ1 µ2 = 1,
m m-1

Q=
j =0

µ

j m-j 1 µ2

-
j =0

µ

j m-j -1 1 µ2

=

m j =0

µ2j - 1 µ

m-1 j =0 m 1

µ

2j +1 1

.

µ1 , Q = (-1)
m m j =0

|µ1 |2j + m-1 |µ1 |2j j =0 |µ1 |m

+1

.

m Q < 0, Q > 0. z 1. µ1 1 µ2 > 0 Q=
m j =0

µ2j - 1 µ

m-1 j =0 m 1

µ

2j +1 1

=

1+

m j =1

µ2j -1 (µ1 - 1)) 1 > 0. µm 1

, , (-1, 1). z = cos . µ1,2 = e±i Q= (µ1 - 1)µm + (1 - µ2 )µ 1 µ1 - µ2
m 2

=

(ei - 1)eim + (1 - e ei - e-i

-i

)e-

im

=

=

(cos - 1 + i sin )(cos (m) + i sin (m)) + (1 - cos + i sin )(cos (m) - i sin (m)) 2i sin
+1) cos (2m2 sin ((m + 1)) - sin (m) 2i((cos - 1) sin (m) + cos (m) sin ) = = . 2i sin sin cos 2

=

2k Q m = 2m+1 , k = 0, . . . , m - 1. , +1 2k z , , Pm (z ) zk,m = cos 2m+1 , k = 0, . . . , m - +1 1.

4. R= (µ1 - 1)µm + (1 - µ2 )µm 2 1 , µ µ1 - µ2
1,2

=w±

w2 - d,

Tm (w) m d. . , d = 1 3. d = 1. [m] 2 R=
j =0

m+1 w 2j + 1

[
m-2j

m-1 2

]

( w - d) -
j =0

2

j

m w 2j + 1

m-2j -1

(w2 - d)j = Tm (w).

R w m, w. Bj (m) Tm (w) = m Bj (m)wj j =0 105

m. m

] m j B (-d) 2l+1 (m) = - 2j + 1 j + l + 1 - m 2 j =0 m 2 m m+1 j B (m) = -l 2 , 2l m (-d) 2j + 1 j + l - 2 j =0 [

m-1 2

m 2

-l-1

, (8)

m B 2l+1 (m) = B (m) = - 2l

[m] 2
j =0
m-1 2

m+1 2j + 1 m 2j + 1

j j+l- j j+l-

m-1 2

(-d)

m-1 2

-l

, (9)

j =0

m-1 2

(-d)

m-1 2

-l

.

3 , Tm (w). w - d. µ2 - d µ1 < 0. R , , µ1 µ2 = d,
m m-1

R=
j =0

µ

j m-j 1 µ2

-
j =0

µ

j m-j -1 1 µ2

= (-1)

m

m j =0

dm-j |µ1 |2j + m-1 d j =0 m |µ1 |

m-1-j

|µ1 |2j

+1

.

m R < 0, R > 0. w d. µ1 d µ2 > 0 R=
m j =0

d

m-j

µ2j - 1 µ
m 1

m-1 j =0

d

m-1-j

µ

2j +1 1

=

dm +

m j =1

d

m-j

µ

2j -1 1

(µ1 - 1)

µ

m 1

. d, =

, µ1 1 , R . d > 1, µ 1 d > 1 R > 0. d < 1, w 1 µ1 > w 1 R > 0. 1 > w µ1 1, w d+1 . , Tm (w) d+1 2 - d, 2 . < - d w < d. w d cos . µ1,2 = de±i i m-1 ( de - 1)eim + (1 - de-i )e-im R=d 2 = ei - e-i m-1 ( d cos - 1 + i d sin )(cos (m) + i sin (m)) =d 2 + 2i sin m-1 (1 - d cos + i d sin )(cos (m) - i sin (m)) +d 2 = 2i sin m-1 ( m-1 d cos - 1) sin (m) + d cos (m) sin d sin ((m + 1)) - sin (m) =d 2 =d 2 . sin sin 106

..

, . g () = d sin ((m + 1)) - sin (m) k k = m+1 , k = 1, . . . , m, k : k = 2l, g (2l ) > 0, k = 2l + 1 (2l+1 ) < 0. , (k , k+1 ), k = 1, . . . , m - 1, g () . , Tm (w) m - 1 . , w = d cos m R , w = d+1 +1 2 . , . , w = d µ1,2 = d m-1 m-1 - md 2 = d 2 ((m + 1) d - m). , d mm , +1 d+1 m = d, ( d cos m+1 , d), d, 2 . m+1 R = (m + 1)d
m 2

, , Tm (w) m , .

3

x Ex . , : p = q p = q . 1. x , n x . , p = q , r 1 (n - x + 1) , k (k = 0, . . . , n - x) (n - k + 1) 2j (-x2 + (2n + 1)x + 2(n + 1))(1 - cos 2n+1 ), +3 j = k , . . . , n. . (. [14]) x Ex r 1. (. (5)) x , .. E sx . t x Ex s e- Ex . p = q r = 1 - 2p. , r 1 p 0. (1) Ex = 1 2 , c = . 2 + (2n + 1)x + 2(n + 1) cp -x
1

s e-cpt . limr 1,2 = 1 - rs ±

1,2 . (1 - s)(1 - s + 4ps) = 2ps
-cpt

1 - s + 2ps ± (1 - rs)2 - 4p2 s2 = 2ps
cpt

=

cpt + o(p) + 2pe-

±

(cpt + o(p))(cpt + o(p) + 4pe 2pe-cpt 107

)

.

, lim
1,2

r 1

=1+

z = 1 + ct , limr1 1,2 2 (5). 2 3 (5) lim Ux (e
-
t Ex

ct(ct + 4) ct ± . 2 2 = z ± z 2 - 1.

r1

Pn-x 1 + )= Pn+1 1 +
j,n+1

ct 2 ct 2

=

n-x-1 (ct + j,n-x j =0 n j =0 (ct + j,n+1 ) 2j +1 2n+3

)

,

(10)

j,n-x = 2(1 - cos

2j +1 2(n-x)+1

),

= 2(1 - cos
m-1

). ) (11)

Sm =
j =0

(ct +

j,n-x

,
m i- 1

Sm =
i=0

Mi (m)
j =0

(ct + j,n+1 ).

(, i = 0 i-1 = 1.) j =0 . m = 1, (11) M1 (1) = 1 M0 (1) = 0,n-x - 0,n+1 : S1 = ct + 0
,n-x

= ct +

0,n+1

+

0,n-x

- 0

,n+1

.

m - 1 . (11)
m-2

Sm = (ct +
m-1

m-1,n-x

)
j =0

(ct +

j,n-x

) = (ct + m

-1,n-x

)Sm

-1

=

i-1

=
i=0

Mi (m - 1)((ct +

i,n+1

) + (m

-1,n-x

-

i,n+1

))
j =0

(ct + j,n+1 ) =

m

i- 1

=
i=0

Mi (m)
j =0

(ct +

j,n+1

).

, ,
m-1 m i-1

(ct + j,n-x ) =
j =0 i=0

Mi (m)
j =0

(ct + j,n+1 ),

Mi (m) Mi (m) = Mi-1 (m - 1) + Mi (m - 1)(m
-1,n-x

- i,n+1 ), i = 1, . . . , m - 1,
m-1,n-x

Mm (m) = 1 M0 (m) = M0 (m - 1)( 108

-

0,n+1

).

..

, (10) P ct 1+ 2
n-x-1 n-x i-1

n-x

=
j =0

(ct +

j,n-x

)=
i=0

Li (n - x)
j =0

(ct +

j,n+1

),

Li (n - x) = Mi (n - x), i = 0, . . . , n - x - 1. Li (n-x) k j . k,n-x j,n+1 , ..
2j +1 2n+3 k,n-x

k,n-x

-

j,n+1

,

-

j,n+1

= 2 cos

2k + 1 2j + 1 - cos 2n + 3 2(n - x) + 1 - x) > 0.

> 0,

2k+1 . , Li (n 2(n-x)+1 n-x-1 , i=0 i,n-x = n=0 i,n+1 = i Pn-x 1 + ct . 2

2k+1 2(n+1)+1

<

1. (7)

P

n-x

ct 1+ 2

[ =2
-n+x

n-x 2

]

j =0

n-x+1 (ct + 2) 2j + 1

n-x-2j

(ct)j (ct + 4)j -

[ -2
1-n+x

n-x-1 2

]

j =0 - n=0x-1 i,n-x = i , n-x+1 1

n-x (ct + 2) 2j + 1
n-x 1

n-x-2j -1

(ct)j (ct + 4)j .

-

= 1.
i-1 n j =i

n-x r1

lim Ux (e

-

t Ex

)=
i=0

L i ( n - x)
j =0

j,n+1

n j =i

j,n+1 . (ct + j,n+1 )

(12)

i - , Li (n - x) > 0, j,n+1 > 0 n=0x Li (n - x) j-1 j,n+1 = 1. , =0 i 0 < Li (n - x) i-1 j,n+1 < 1 j =0 r 1 (n - x + 1) . , i- (i = 0, . . . , n - x), n - i + 1 2j 2 (1 - cos 2n+1 ), j = i, . . . , n, c +3 Li (n - x) i-1 j,n+1 . j =0

2. p = q , r 1 x n x 1) 1, q p-1 0; 2) (x + 1) (x + 1), q p-1 ; 3)
n+1

p(u) =
k=1

^ Tn-x (k,n+1 ) k e Hk (k,n+1 ) 109

,n+1

u

,

q p-1 d = 0 1. k,n+1 , k = 1, . . . , n + 1, Hk (t) =

^ Tn+1 (u) = 0, k
m -m j =0 k=j m

,n+1

< 0,

^ Tn+1 (t) ^ Tm (t) = d t - k,n+1

Bk (m)

k (ct)j (d + 1) j
x-n -1

k-j -k

2,

c = (d - 1)2 ((d - 1)(x + 1) + d Bk (m) (8), (9).

-n-1

-d

),

(13)

. 1. r p q , , p + q 0. , , p q , r 1 p q 0. p 0 q 0. 1 p = q : Ex = 1 x+1 +q q ( p - 1)p ( p - 1)2 p q p
-n-1

-

q p

x- n

.

q , Ex p . : q p-1 0, q p-1 , q p-1 const = 0 1. 1. q p-1 0. 1, , t e- Ex s. , A B A ( A B ), lim B = 1. q p-1 0 Ex

Ex

x+1 1q - + p pp

-n-1

1 - (x + 1) = p
q p

q p

n+1

n+1

.

-

e
1,2

s 1,2 1 - s + (p + q )s ± (1 - s + ( p + q )2 s)(1 - s + ( p - q )2 s) = = 2ps
pt
q (p) qn +1)( p ) n+1

q n+1 pt p q n+1 1-(x+1) p

() ()

-
+1

=

1-(x

+ o(p) + (p + q )e
-
q n+1 pt p q n+1 1-(x+1) p

q n+1 pt p q n+1 1-(x+1) p

() ()

2pe

q pt p 1-(x

() ()

±

() qn +1)( p )

n+1

- +1

+o(p)+( p+ q )2 e

q n+1 pt p q n+1 1-(x+1) p

() ()

1-

() q (x+1)( p )

q pt p

n+1 n+1

-

+o(p)+( p- q )2 e

q n+1 pt p q n+1 1-(x+1) p

() ()

±
-

2pe

q n+1 pt p q n+1 1-(x+1) p

() () 110

.

..

1+ , , (5),
r1 1 1,2 q p

+t

q p

n+1

± 1-

q p

1 + 2t

1+

(1

-

q p q2 p

)

q p

n+1

2 1+t q p
n+1

.

,

2

q p

lim Ux (e

-

t Ex

q ) = lim r1 p

x+1

t t

q p q p

n+1

1+t
n+1

q p q p

n+1

n-x

+ 1-
n+1 n+1

q p q p

q p q p

n-x

=

n+1

1+t
x+1

+ 1-
n+1 n-x

q = lim r 1 p
x+1 q p

n-x

= 1 . t+1

1-
q p n+1

q p

+t
q p

q p

1+t
q p

q p n+1

n+1

1-

+t 1+t

, 1. 2. q p-1 pq -1 0. (2), p , q x+1 + q (1 - p ) (1 - p )2 q q q
p q

Ex =
1,2

p q

n+1

-

p q

n-x

x+1- q

p q

n-x+1

.

=

1 - s + (p + q )s ±

(1 - s + ( p + q )2 s)(1 - s + ( p - q )2 s) = 2ps
-

qt

=

x+1-

(p) q

n-x+1

+ o(q ) + (p + q )e
-

qt p n-x+1 x+1- q

()

2pe

qt p n-x+1 x+1- q -

qt p n-x+1 x+1- q

±
qt p n-x+1 x+1- q

()

±

()

+o(q )+( p+ q )2 e

qt p n-x+1 x+1- q

qt p n-x+1 x+1- q

()

-

()

+o(q )+( p- q )2 e

()

-

2pe

qt p n-x+1 x+1- q

.

()

1+ 1 ,
1 ,2 q p

1+

t x+1

±

q p

1+

t x+1

1+2 2

t x+1

-1

1+

-2 p t x+1 q

.

q t 1 (1 + ), lim 2 = r 1 p x+1 1+ x 111

t +1

.

,
r1

lim Ux (e-
q p q p

t Ex

)= -1 + 1-
1
t x+1

q = lim r 1 p
x+1

q p q p

1+ 1+

t x+1 t x+1 x+1

n-x

1+ 1+
t x+1

t x+1 t x+1

1 +1
t x+1

n-x +1 n+1 +1

=

n+1

q = lim r1 p

q p q p

1+

-1 + 1- n-x+1
n+2

1
t x+1

1 +1
t x+1

1+

t x+1

=

1 1+
x+1 t x+1

.

, (x + 1) x + 1. 3. q p-1 d = 0 1. Ex (x + 1)(d - 1) + d-n-1 - dx (d - 1)2 p
-n

1 (d - 1)2 = , c = cp (d - 1)(x + 1) + d-

n-1

- dx

-n

.

, 1,2 , r 1 - p - q ,
1,2

=

1 - s + (p + q )s ±

(1 - s + ( p + q )2 s)(1 - s + ( p - q )2 s) . 2ps

e-cpt ,
1,2

=

cpt + o(p) + (p + q )e- 2pe-cpt

cpt

±

±

(cpt + o(p) + ( p + q )2 e-cpt )(cpt + o(p) + ( p - q )2 e-cpt ) . 2pe-cpt d + 1 + ct = ± 2
1

(ct + (1 +

d)2 )(ct + (1 - d)2 ) 2 .

r1

lim 1

,2

w =

d+1+ct 2

, limr
-
t Ex

1,2 = w ±
x+1 n-x k=0 n+1 k=0

w2 - d. 4
d+1+ct k 2 d+1+ct k 2

r 1

lim Ux (e

)=d

Bk (n - x) Bk (n + 1)

=

^ Tn-x (t) , ^ Tn+1 (t)

^ Tm (t) = Tm d + 1 + ct 2 c= (d - 1)2 (d - 1)(x + 1) + d-n-1 - d
x-n

.

, . [3], ^ Tn-x (t) , , ^ T (t)
n+1

n+1

k=1

^ Tn-x (k,n+1 ) e Hk (k,n+1 )

k

,n+1

u

,

k

,n+1

^ - Tn+1 (t) = 0,

112

..

Hk (t) =

^ Tn+1 (t) ^ Tm (t) = d t - k,n+1

m -m

m

Bk (m)
j =0 k=j

k (ct)j (d + 1) j

k-j -k

2.

4 , Bm (m) = 2m . cm ^ ^ ^ Tm (t) d-m Bm (m)cm 2-m = d Tm (t) Tm (t) = n+1 c j =1 ( d t - j ). Hk (t)
n+1

Hk (t) =
j =1, j =k

c t - j,n+1 . d
d+1 2

Tm (w) - d, ^ k Tm (t) = Tm d+1+ct . 2

, ,

q p-1 d = 0. . n = 1. , . x = 0 (5) U0 (s) = d limr
1

2 - 1 - 2 + 2 1 2 . 3 - 2 - 3 + 2 1 1 2 2
d+1+ct 2

1,2

=w±

w2 - d w =
-
t E0

,

r1

lim U0 (e

)=d

(

ct)2

ct + d = + (2d + 1)ct + d2
2d+1- 4d+1 2d+1+ 4d+1 2 2 2d+1- 4d+1 ct + 2d+1+2 4d+1 2

2d + 1 - 4d + 1 2d+1+2 4d+1 4d + 1 - 1 = + 2d+1+ 4d+1 2d 2d ct + ct + 2

.

(13), d+1 4d + 1) 2 (2d + 1 + 2d d+1 4d + 1) d+1 (2d + 1 + 4d + 1). 2 (2d + 1 - 2 2d 2d d2 x = n = 1 c = 2d+1 U1 (s) = d2 . 2 + 1 2 + 2 - 1 - 2 1 2

lim U1 (e
-
t E1

r1

)= ct +

2d+1- 4d+1 2d+1+ 4d+1 2 2 2d+1- 4d+1 ct + 2d+1+2 4d+1 2

.

, d+1 d+1 22d2 (2d + 1 - 4d + 1) 22d2 (2d + 1 + 4d + 1). 113

4

, 2, , i- , Dx (i) = max(Sx (i - 1) + 1 - zi - n, 0). , , x = min(i : Sx (i) < 0), x n x
x

mx (n) = E
i=1

v i Dx (i),

0 < v < 1

.

5. x n x mx (n), , mx (n) = a1 , a
2

a

x+1 2

-a n

x+1 1

,

v pa2 + (v r - 1)a + v q = 0, n = a
n+1 2

(a2 - 1) + an+1 (1 - a1 ). 1

. 1, mx (n) = v (pmx+1 (n) + rmx (n) + q mx-1 (n)) 1 x n - 1 m0 (n) = v (pm1 (n) + rm0 (n)), mn (n) = v ((p + r)mn (n) + q mn-1 (n) + p).

1. v pa2 + (v r - 1)a + v q 0 1, : 0 < a1 < 1, a2 > 1. 3. , , , , (1 - a1 )2 a n = log a2 - 1, 2 1 (a2 - 1) n x, , n < x. . mx (n) n mx (n + 1) - mx (n) = (a (14) 0 n = log
a2 a1

x+1 2

-a

x+1 1

)

n - n+1 . n n+1

(14)

(1 - a1 )2 - 1. (a2 - 1)2 114

..

n mx (n) 0, ..
n

lim (mx (n + 1) - mx (n)) = lim

n

-(a

x+1 2

-a

x+1 1

)

an+1 (a2 - 1)2 2 2 a2n+3 (a2 - 1)2

=0

ax+1 - ax+1 2 1 = 0. n+1 n n a2 (a2 - 1) , 0 . , , , . lim mx (n) = lim
- 1. p < q 1 + 1vqv , x. 2

2. , , .. n x, ax+1 (a2 - 1)2 - ax+1 (1 - a1 )2 0. 2 1 , , , y n (y ). i- Sx,y (i) = min(Sx,y (i - 1) + 1 - zi , n), Sx,y (i - 1) + 1 - zi 0, y , Sx,y (i - 1) + 1 - zi < 0.

, Sx,y (0) = x, n x y n

mx (n, y ) = E
i=1

v i Dx,y (i),

v

, Dx,y (i) = max(Sx,y (i - 1) + 1 - zi - n, 0).

6. x, y n x, y mx (n, y ) c1 (y )ax + c2 (y )ax 2 1 , mx (n, y ) = n (y ) a1 , a
2

v pa2 + (v r - 1)a + v q = 0,

c1 (y ) = q ay - pa1 , c2 (y ) = pa2 - q ay , n (y ) = an (a2 - 1)c2 (y ) - an (1 - a1 )c1 (y ). 2 1 2 1 . mx (n, y ) = v (pm m0 (n, y ) = v (pm1 (n, y ) + rm0 (n, y ) + q my (n, y )), mn (n, y ) = v ((p + r)mn (n, y ) + q mn-1 (n, y ) + p).
x+1

(n, y ) + rmx (n, y ) + q mx-1 (n, y )) 1 x n - 1

115

. 4. , , max(x, y ). . mx (n + 1, y ) - mx (n, y ) = (c1 (y )ax + c2 (y )ax ) 2 1 , q ay > q a-1 = pa1 , pa2 = q a-1 > q ay . 1 2 2 1 C, c1 (y ) > 0 c2 (y ) > 0. , n (y ) - n+1 (y ) = -an (a2 - 1)2 c2 (y ) - an (1 - a1 )2 c1 (y ) < 0. 2 1 , an (a2 - 1)c2 (y ) = an (a2 - 1)q a-1 (1 - ay+1 ) = 1 2 2 1 = q (a1 a2 )-1 (1 - a1 )(a2 - 1)an+1 (1 + a1 + ... + ay ) > 1 2 -1 n+1 > q (a1 a2 ) (1 - a1 )(a2 - 1)a1 (1 + a2 + ... + ay ) = 2 = an (1 - a1 )q a-1 (a 1 2
y +1 2

n (y ) - n+1 (y ) . n (y )n+1 (y )

- 1) = an (1 - a1 )c1 (y ). 1

, n y n (y ) > 0, , n+1 (y ) > 0. , mx (n, y ) , mx (n, y ) n, , n. , x n y n, n = max(x, y ). 5. x , , x. . 4 y x n = x. mx (x, y ) = y : mx (x, y + 1) - mx (x, y ) = c1 (y + 1)ax + c2 (y + 1)ax c1 (y )ax + c2 (y )ax 1 2 1 2 - . x (y + 1) x (y ) c1 (y )ax + c2 (y )ax 1 2 . x (y )

4 , x (y ) > 0 x y , , , . (c1 (y + 1)ax + c2 (y + 1)ax )(ax (a2 - 1)c2 (y ) - ax (1 - a1 )c1 (y ))- 2 2 1 1 -(c1 (y )ax + c2 (y )ax )(ax (a2 - 1)c2 (y + 1) - ax (1 - a1 )c1 (y + 1)) = 1 2 2 1 = (a1 a2 )x (a2 - a1 )(c1 (y + 1)c2 (y ) - c1 (y )c2 (y + 1)) = = q 2 ( a1 a2 )
x-1

(a2 - a1 )((a

y +2 2

- 1)(1 - a
y +1 2

y +1 1

) - (a

= q (a1 a2 )x-1 (a2 - a1 )(a2 - 1)(1 - a1 )[a

(1 + ... +

y +1 2 ay ) 1

- 1)(1 - a -a
y +1 1

y +2 1

)) =

(1 + ... + ay )]. 2

, .. , y , , , , y = x. 116

..

6. n = y = x . . n = y = x mx (x, x) = mx (x, x) = a
x+1 2

c1 (x)ax + c2 (x)a 1 x (x)

x 2

ax+1 - ax+1 2 1 x+1 (a2 - 1) + a1 (1 - a1 ) - (a2 - a1 )(a1 a2 )x

+1

.

, x (x) > 0, (ax+2 - ax+2 )(ax+1 (a2 - 1) + ax+1 (1 - a1 ) - (a2 - a1 )(a1 a2 )x+1 )- 2 1 2 1 -(ax+1 - ax+1 )(ax+2 (a2 - 1) + ax+2 (1 - a1 ) - (a2 - a1 )(a1 a2 )x+2 ) = 2 1 2 1 = (a2 - a1 )(a1 a2 )x+1 (a2 - 1)(1 - a1 )[(1 + ... + ax+1 ) - (1 + ... + ax+1 )]. 1 2 , a1 < 1 < a2 , , , , x = 0.

5

. , , , , .. . : , . i , n, . , Sx (i - 1) + 1 - zi - n, Sx (i - 1) + 1 - zi n, Sx (i - 1) + 1 - zi , Sx (i - 1) + 1 - zi < 0, Lx (i) = 0, 0 Sx (i - 1) + 1 - zi n.
x

x (n) = E
i=1

v i Lx (i),

v < 1

.

7. x n x (an+1 (1 - a2 ) - 1)ax+1 - (an+1 (1 - a1 ) - 1)ax+1 1 1 2 . x (n) = 2 n . x (n) = v (px+1 (n) + rx (n) + q x-1 (n)) 1 x n - 1 117

0 (n) = v (p1 (n) + r0 (n) - q ), n (n) = v ((p + r)n (n) + q n-1 (n) + p).

, . 7. , . . x (n) n , n . x (n + 1) - x (n) = ((a
n+1 2

((a

n+2 2

(1 - a2 ) - 1)ax 1
+1

+1

- (an+2 (1 - a1 ) - 1)a 1 n n+1
x+1 2

x+1 2

)n

-

-

(1 - a2 ) - 1)ax 1

- (an+1 (1 - a1 ) - 1)a 1 n n+1
x+1 2

)n+1
+1

=

=

(1 - a1 )(1 - a2 )(a2 - a1 )(ax+1 a2 - a1 a 1 n n+1 + (ax 2
+1

)(a1 a2 )n

+

-a

x+1 1

)(a

n+1 1

(1 - a2 ) - a 1 n n+1

n+1 2

(1 - a2 )) 2
n+1 2

.

n n = a

n+1 2

(a2 - 1) + a

n+1 1

(1 - a1 ) a
x+1 1

(a2 - 1).

n

lim x (n) = a

n

lim (x (n + 1) - x (n)) = lim

n

-

an+1 (1 - a2 ) 2 2 2 a2n+3 (a2 - 1)2

= 0.

, n. y , Lx,y (i) : Sx,y (i - 1) + 1 - zi - n, Sx,y (i - 1) + 1 - zi n, Sx,y (i - 1) + 1 - zi - y , Sx,y (i - 1) + 1 - zi < 0, Lx,y (i) = 0, 0 Sx,y (i - 1) + 1 - zi n. C

x (n, y ) = E
i=1

viL

x,y

(i).

118

..

8. x, n x y n x (n, y ) = c1 (y )ax + c2 (y )ax - q (y + 1) (x, n) 1 2 , n (y )

(x, n) = (a2 - 1)an ax + (1 - a1 )an ax . 12 21 . x (n, y ) = v (p 0 (n, y ) = v (p1 (n, y ) + r0 (n, y ) + q y (n, y ) - q (y + 1)), n (n, y ) = v ((p + r)n (n, y ) + q n-1 (n, y ) + p).
x+1

(n, y ) + rx (n, y ) + q x-1 (n, y )) 1 x n - 1

. 8. , . . (c1 (y )ax + c2 (y )ax )(n (y ) - n+1 (y )) 1 2 x (n + 1, y ) - x (n, y ) = - n (y )n+1 (y ) - q (y + 1)((a2 - 1)an+1 ax + (1 - a1 )a 2 1 n+1 (y ) =- -
n+1 x a2 1

)

+

q (y + 1)((a2 - 1)an ax + (1 - a1 )an ax ) 12 21 = n (y )

(c1 (y )ax + c2 (y )ax )(an (a2 - 1)2 c2 (y ) + an (1 - a1 )2 c1 (y )) 1 1 2 2 - n+1 n n (a2 (a2 - 1)c2 (y ) - a1 (1 - a1 )c1 (y ))(a2 (a2 - 1)c2 (y ) - an+1 (1 - a1 )c1 (y )) 1

q (y + 1)(a2 - 1)(a1 a2 )n (1 - a1 )(a2 - a1 )(c1 (y )ax + c2 (y )ax ) 2 1 . n+1 n+1 n n (a2 (a2 - 1)c2 (y ) - a1 (1 - a1 )c1 (y ))(a2 (a2 - 1)c2 (y ) - a1 (1 - a1 )c1 (y )) c1 (y )ax + c2 (y )a 1 c2 (y )an+1 2
x 2

n lim (x (n + 1, y ) - x (n, y )) = lim - = 0.

n

n

C, . lim x (n, y ) = - q (y + 1)ax 1 . c2 (y )

n

, x (n, y ) n. .. . 119

[1] Albrecher H., Thonhauser S., Optimality results for dividend problems in insurance // RACSAM Revista de la Real Academia de Ciencias, Serie A, Matematicas, 2009, v. 103, 2, p. 295320 [2] Avanzi B., Strategies for dividend distribution: a review // North American Actuarial Journal, 2009, v. 13, 2, p. 217251. [3] Bateman H., Erdelyi A., Tables of integral transforms. McGraw Hill. Book Company, 1959. [4] Bulhmann H., Mathematical methods in risk theory. Springer Verlag, Berlin, 1970. [5] De Finetti B., Su un'impostazione alternativa della teoria colletiva del rischio // Transactions of the XVth International Congress of Actuaries, 1957, v. 2, p. 433443. [6] Dickson D. C. M.,Waters H. R., Some optimal dividend problems // Astin Bulletin, 2004, v. 34, 1, p. 4974. [7] Gerber H. U., An introduction to mathematical risk theory. Huebner Foundation, Philadelphia, 1979. [8] Lundberg F., Approximation of the probability function. Reinsurance of collective risks // Doctoral thesis, 1903. [9] Wenguang Yu, Randomized dividends in a discrete insurance risk model with stochastic premium income // Mathematical Problems in Engineering, 2013, v. 2013, Article ID 579534, 9 pages [10] Wu X., Li S., On a discrete time risk model with time-delayed claims and a constant dividend barrier // Insurance Markets and Companies: Analysis and Actuarial Computations, 2012, v. 3, 1, p. 5057. [11] . ., , // . . -. . 1, . , 2012, 2, p. 5457. [12] - ., . ., // , 1995, v. 20, p. 257277. [13] ., , .1. , , 1984. [14] ., , .2. , , 1984. [15] . ., // . . -. . 1, . , 2009, 5, p. 6062. [16] . ., // , 2010, . 17, 6, . 830838.

120

..

1
..2

, . , , . . .

1

[1, 2] . , [3, 4], , . [5] . [6, 7, 8]. , . ( ). , [9] , [10, 11] . . [12] , . , , , ( ) rk , r (0, 1), k . , , . , , , ,
14-01-00075. , avlebed@yandex.ru, , , . .. .
2 1

121

, . = (1 - a )/(1 - a /µ) (0, 1) a (0, 1) , > 0 , µ > 1 . , [12] . , . , [3, . 6, §28.2], , . 2 3 , , n- Mn Zn > 0 n . 4 Mn Z0 = n . 2 , ( ) , , ( ) 0 rk < 1, k . , ( ) rk , k 1, , . 2.1 ( 2 ). , ( , ). , c [13] " ". [14] . [15] . 2.2 ( ). g (n), , g (0) = 1 g (n + 1) 1, g (n) g (n) , n .

, n- ( ) µn = g (n + 1)/g (n), n 0. , g (n) C n , C > 0, > 0, g (n) exp{cn }, c > 0, 0 < < 1, n . µn - 1 /n µn - 1 c/n1- , n , . , 2.3 ( µ 2 ), . , [12] ( ). , . 122

..

3 , ( ) , - , , . ( ), , . [16, §5.4] ( ). , . , , . , . . = 1. , . , -, , . 4 , . , Z0 = n, n .

2

2.1. . T (n, k ) n- , k , R(n, k ) n- , k , n k . , [4, §2.6], EZn (Zn - 1) = DZn = n 2 . B = 2 /2. 1. E(R(n, k )|Zn > 0) = B B 2 n, P(Zn > 0) n . (1)

. k , , (n - k )- . ET (n, k ) = EZn-k EZk (Zk - 1)/2 = k 2 /2 = k B . ER(n, k ) = ET (n, k ) - ET (n, k - 1) = B . R(n, k ) = 0 Zn = 0, E(R(n, k )|Zn > 0) = B /P(Zn > 0). [4, 10], P(Zn > 0) 1/(B n), n , E(R(n, k )|Zn > 0) B 2 n, n . , : , 1 n, . , (x) = exp{-e-x } . 123

, s (a(s)x + b(s)) (x), s , 1 a(s) = (2 ln s) , b(s) = (2 ln s)1/2 - 2 (2 ln s)-1/2 (ln ln s + ln 4 ),
-1/2

(2)

(x) . ¯ un = a(n)x + b(n), n 1. n(un ) e-x , (un )n (x), n . 1. sup rk = < 1 rk ln k 0, k . P(Mn un |Zn > 0) 1 + B e
-x -1

,

n .

(3)

. X1 , . . . , XN ij , |ij | < 1. [17, 4.2.4] P
1iN

Xi u

- N (u) K ( )
1i
|ij | exp -

u2 1 + |ij |

.

(4)

, 1, |P(Mn un |Zn > 0) - E((un )Zn |Zn > 0)| n u2 K ( )E R(n, k )rk exp - n Zn > 0 1 + rk k=1
n

=

= K ( )
k=1

E(R(n, k )|Zn > 0)rk exp -
n

u2 n 1 + rk

=

(5)

=

K ( )B P(Zn > 0)

rk exp -
k=1 n

u2 n 1 + rk ,

n .

K ( )B 2 n
k=1

rk exp -

u2 n 1 + rk

¯ [17, 4.3.2] 1 n(un ), n 1,
n

n
k=1

rk exp -

u2 n 1 + rk

0,

n ,

(6)

(5) |P(Mn un |Zn > 0) - E((un )Zn |Zn > 0)| 0, , [4, 10], Zn Zn > 0 Bn E((un )Zn |Zn > 0) E(x) (7) (8) . 124
B

n .

(7)

- Exp(1),

d

n ,
-x -1

= 1 + Be

,

n .

(8)

..

, . , (6), (3), [17, §4.3], , , rk , k 1, - . , . 2.2. . , Zn n 1 ( ) EZn = g (n). Zn d - Exp(1), g (n) n . (9)

, µ, E ( - 1) = 2D = 2µ(µ - 1). 2. ET (n, k ) = g (n)(g (n) - g (n - k )) , g (n - k ) (10)

. Zn,k n- , (n - k )- . Zn,k g (n)/g (n - k ). , ET (n, k ) = EZn-k EZn,k (Zn,k - 1)/2 = g (n) g (n) = g (n - k ) -1 , g (n - k ) g (n - k )

(10). , ET (n, n) = EZn (Zn - 1)/2 g (n)2 , n . (11) 2. s (0, 1) ms (n), n 1, , 1 ms (n) n g (n - m(n)) 1, g (n)s ms (n) , n .

. m(n) = 1 g (n) = 1 m(n) = min{1 k n : g (n - m(n)) < g (n)s } g (n) > 1, n. , nk , k 1, ms (nk ) < M , g (n - M ) < g (n)s , g (n)/g (n - M ) 1, . , ms (n) , n . g (n - ms (n))/g (n)s < 1 g (n - ms (n) + 1)/g (n)s 1. g (n - ms (n) + 1)/g (n - ms (n)) 1, g (n - ms (n))/g (n)s 1, n . 125

2 3 ET (n, ms (n)) g (n)
2-s

,

n .

(12)

(9), un = a(g (n))x + b(g (n)), n 1. , exp{-u2 } e-2x g (n)-2 ln g (n), n . (13) n 3. sup rk = < 1 rms (k) ln g (k ) 0, k , 2 /(1 + ) < s < 1, -1 , n . P(Mn un ) 1 + e-x . (4), 1. n = supk>ms (n) rk , n = o((ln g (n))-1 ) g (n)n 1, n . : 1 ms (n) ms (n) + 1 n. (11), (12) (13)
n

|P(Mn un ) - E(un ) | K ( )
k=1

Zn

ER(n, k )rk exp -

u2 n 1 + rk

K ( ) ET (n, ms (n)) exp -

u2 u2 n + K ( )n ET (n, n) exp - n = 1+ 1 + n = g (n)2-s g (n)(-2/(1+))(1+o(1)) + o((ln g (n))-1 )g (n)2 g (n)-2/(1+n ) (ln g (n))1/(1+n ) = = g (n)(2/(1+)-s)(1+o(1)) + o (ln g (n))-n (1+o(1)) g (n)2n (1+o(1)) 0, n .

(9),
n

lim P(Mn un ) = lim E(un )Zn = E(x) = 1 + e-
n

x -1

.

2.1. g (n) C n , C > 0, > 0, ms (n) n - C rk ln k 0, k . , 1 ( ), . 2.2. g (n) exp{cn }, c > 0, 0 < < 1, ms (n) (1 - s rk = o k
- 1/ (s-1)/ s

n n,

ln g (n) ln n,

n ,

)n,

ln g (n) cn , k .

n ,

,

. 2.3. . , [3, . 1, §8]: Zn a.s. - W, µn 126 n , (14)

..

- (s) = Ee- (µs) = f ((s)),

sW

-

f . q s = f (s) [0, 1). q < 1 p = 1 - q = P(W > 0) > 0. P(Zn > 0) p, n . 3. E(T (n, k )|Zn > 0) = µn
-1

µk - 1 2 + µ2 - µ µ - 1 2P(Zn > 0)

(15)

. 1, ET (n, k ) = EZn-k EZk (Zk - 1)/2. , EZn-k = µn-k . Zn [3, . 1, §5], EZk (Zk - 1) = µk - 1 2 + µ2 - µ . µ-1 2 T (n, k ) = 0 Zn = 0, E(T (n, k )|Zn > 0) = ET (n, k )/P(Zn > 0), (15). ET (n, k ) = µ
n-1

2 µk (µk - 1) +µ µ2 - µ

2k

- µk = ( 2 + µ2 - µ)µ

k -1

µk - 1 , µ-1

4 , E(T (n, n)|Zn > 0) C (µ, , p)µ2n , ms (n) = [(1 - s)n], 0 < s < 1, E(T (n, ms (n)|Zn > 0) µ
(2-s)n+O(1)

n ,

(16)

n ,

(17)

(14), un = a(µn )x + b(µn ), n 1. , exp{-u2 } e n
-2x -2n

µ

(n ln µ),

n .

(18)

4. sup rk = < 1 rk = o(1/k ), k , P(Mn un |Zn > 0) E(exp{-e-x W }|W > 0), n . (19)

. (4), 1. 2 /(1 + ) < s < 1. n = supk>ms (n) rk , n = o(1/n) nn 1, n . : 1 ms (n) ms (n) + 1 n. (16), (17) (18) |P(Mn un |Zn > 0) - E((un )Zn |Zn > 0)| n u2 K ( ) E(R(n, k )|Zn > 0)rk exp - n 1 + rk k=1 u2 K ( ) E(T (n, ms (n)|Zn > 0) exp - n + 1+ u2 +K ( )n E(T (n, n)|Zn > 0) exp - n = 1 + n = µ(2-s)n+O(1) µ(-2/(1+))(1+o(1)) + o(1/n)µ2n µ-2n/(1+n ) (n ln µ)1/(1+ = µ(2/(1+)-s)n(1+o(1)) + o(n-n (1+o(1)) )µ2nn (1+o(1)) 0, n . 127

n

)

=

E((un )Zn |Zn > 0) E((x)W |W > 0) = E(exp{-e-x W }|W > 0), n ,

. (19). , -, , , , .

3

m- , 0 < m < m < 0. ¯ , A(x) x- L(x), c > 0 (20)

(i, n, m) i- n- n, 1 i Zn . (i, n, n) = i (i, n, m) = 1 A R+ , x , L , > 0. ¯ ¯ A(x/c) c A(x), x .

¯ v (s) , , sA(v (s)) 1, s . , v (s) 1/ [18, §1.5]. , i- n-

n,i =
k=0

ak

n-k,(i,n,n-k)

,

(21)

n,i , n Z, i 1, A, ak , k 0, ,

a = 1. k
k=0

(22)

(21)

F ( x) =
k=0

A

x ak

,

¯ ¯ (20) (22) F (x) A(x), x , A(x) = exp{-(x/c)- }, x > 0, c > 0 ( ) F = A. . , [16, §5.4] (X, Y ) F G U = lim P(Y > G-1 (t)|X > F
t1-0 -1

(t)),

. U = 2 - lim 1 - P(X F
t1-0

(t), Y G-1 (t)) . 1-t

-1

(23)

128

..

4. , k ,

U (k ) =
j =k

a , j

k 1.

. X Y , (n - k )- , , k , . ,

P(X u, Y u) =
j =k

A

u aj

k-1

2

A
j =0

u aj

k-1

= F (u)
j =0

A

u aj

,

(20)
k-1

1 - P(X u, Y u)

1+
j =0

a

j

¯ A(u),

u .

(23) . , k (0, 1), k 1, , a0 = (1 - 1 )1/ ak = (k - k+1 )1/ , k 1, U (k ) = k k 1. , , - n,i = an-1
,(i,n,n-1) bn,i ,

a (0, 1),

b > 0,

(24)

n,i i- n- , n,i A. b = (1 - a )1/ (21) ak = bak , U (k ) = a k , k 1, 5. Z (m, n) m n, n. Z (m, n), 0 m n [4, §10].

5. K 1 ((Z (n - k , n)/cn )0
k K

|Zn > 0) - (bk )0

d

kK

,

n ,

(25)

, bk 0, k 0, cn > 0, n 1, , un = xv (cn ), n 1, x > 0. P(Mn un |Zn > 0) E exp{-x- }, =
k=0

n ,

(26)

a bk . k

(27)

129

. , (21)

Z (n-k,n)

k,l ,

Mn =
k=0

d

a

k l=1

Z (m, n) = 1 m < 0. 1 K n.
K - Mn = k=0

Z (n-k,n)

k,l ,

a

k l=1

a
k

Z (n-K,n)

K

a
k

Z (n-k,n)

k,l ,

(28)

+ Mn = k=K +1

k,l
l=1 k=0

l=1

- + 3 Mn d Mn d Mn , Z (n - k , n) k . bk , k 1, . T (25) (28) + lim inf P(Mn un |Zn > 0) lim P(Mn un |Zn > 0) = n n K

= E exp - U (K + 1)bK +
k=0 n

a b k
n

k

x-

,

- lim sup P(Mn un |Zn > 0) lim P(Mn un |Zn > 0) = K

= E exp -
k=0

a b k

k

x-

,

K (26). 3.1A. , [19], cn = n, Exp(1/B ), bk = P(Zk > 0) = 1 - f (k) (0), f (k) k - , P(Mn xv (n)|Zn > 0) 1 + B e- =
k=0 x -1

,

n ,

a (1 - f k

(k)

(0)).

(29)

, -, ( ), 1 - f (k) (0) = 1/(k + 1) ln(1 - a ) . = -(1 - a ) a

(30)

a 0 1, a 1 0, . 3.1B. , f (s) = s + (1 - s)1+ L(1 - s), (0, 1], L(t)
d : X d Y , P(X > u) P(Y > u).
3

130

..

[18], [20], bk = P(Zk > 0) = 1 - f 1 - (1 + t- )-1/ , P(Mn xv (cn )|Zn > 0) 1 + x
- -1/

(k )

(0), cn = 1/bn , Ee

-t

=

,

n ,

(29). 3.2. cn = g (n), Exp(1), bk 1. (9) n n - K , , ((Zn-k /m)0
kK

|Zn-K = m) (g (n - k )/g (n - K ))0k

K

,

m ,

g (n - k )/g (n - K ) 1, n , 0 k K . (22) = 1 P(Mn xv (g (n))) 1 + e
-x -1

,

n .

3.3. (14) [21] cn = µn , d = (W |W > 0), bk = µ-k . P(Mn xv (µn )|Zn > 0) E(exp{-x- W }|W > 0), =
k=0

n ,

a µ - k . k

, - = (1 - a )/(1 - a /µ). (31)

, , , (14) , EZ1 ln+ Z1 < , . . [3, . 1, §7.1] - f ( s) = 1 - bs b + , 1 - c 1 - cs b, c > 0, b + c 1,

( ), µ= µ > 1 q= 1 - (b + c) . c(1 - c) b , (1 - c)2

[3, . 1, §8.5] (W |W > 0) p = 1 - q , P(Mn xv (µn )|Zn > 0) 1 + x- /p
-1

,

n .

- ( ). . 1 (a): 3.1A (30) c = 2 , 3.3 (31) = 2, µ = 3 . 131

. 1: (a)

4

Mn Z0 = n . Mn n , Y = =1 Xi , i , X1 , . . . , X . , 0 < µ < , X1 , . . . , X F , ( 2). Y G. = 0, Y = -, G . P(Z1 > 0) 1 n , Mn ( ) , Z1 > 0 ( Mn > -). s (0, 1) un (s), ¯ F (un (s))µn s, nF (un (s)) (- ln s)/µ, Z1 EF (un (s)) s, n , . , P(Mn un (s)) s , > 0. , [17, 1.7.13], 0 < < , ¯ un , nF (un ) , , 1 - F (x) 1, 1 - F (x - 0) x xF = sup{x : F (x) < 1}.

F - - . 132

..

U . 6. U = 0 2 < , = 1. . k (k - 1) ¯ ¯ P(X1 > x, X2 > x) 1 - P(Y > x| = k ) k F (x), k F ( x) - 2 ¯ P(Y > x| = 1) = F (x), k 2,

2 + µ2 - µ ¯ ¯ ¯ µF (x) - P(X1 > x, X2 > x) G(x) µF (x). 2 ¯ - U = 0 P(X1 > x, X2 > x) = o(F (x)), u , ¯ (x) µF (x), x , P(Mn un (s)) = G(un (s))n s, n . ¯ G U = 0 , , < 1. , . [16]. C [0, 1]d , d 2, [0, 1]. , Rd F (x1 , . . . xd ) = C (F1 (x1 ), . . . Fd (xd )), Fi , 1 i d, . , . , .
d

Cd (y1 , . . . , yd ) =

-1 i=1

(yi ) ,

(32)

[0, 1], , (0) = +, (1) = 0. d = 2 , . , -1 , (32) d 2. , , X1 , . . . , X = d 1 (32). 7. (y ) c(1 - y ) , y 1 - 0, = E
1/

/µ.

. 1 - -1 (t) (t/c)1/ , t 0 + 0, 1 - Cd (1 - , . . . , 1 - ) d1/ , Y , ¯ G(un (s)) = 1 - EC (F (un (s)), . . . , F (un (s)) E . 133
1/

0.

(33)

¯ F (un (s)),

n ,

, 1, E 1/ µ < . , 6 > 1, , , - (y ) = (- ln y ) : 1/ d Cd (y1 , . . . , yd ) = exp - (- ln yi ) , 1.
i=1

, , , 5: U = 0. , U . (23) (33) U = 2 - 21/ . = 1/ log2 (2 - U ). , = E log2 (2-U ) /µ, 1 P( > 0)/µ c U 0 1. , , 1 2 p1 p2 , : = (p1 + p2 (2 - U ))/(p1 + 2p2 ). , 6 , . , F ( 2), P(Mn a(µn)x + b(µn)) exp{-e-x }, n ,

a(n) b(n) (2). ¯ F (x) x- L(x), x , L , > 0, ¯ v (r) , , rF (v (r)) 1, r ( 3), P(Mn xv (µn)) exp{-x
-

},

n .

[1] Arnold B.C., Vil lasenor J.A., The tallest man in the world // Statistical theory and applications. Papers in honor of H.A.David. Springer, 1996, p. 8188. [2] Pakes A.G., Extreme order statistics on Galton-Watson trees // Metrika, 1998, v. 47, p. 95117. [3] T., . .: , 1966. [4] .., . . . 8. .: , 2008. [5] Mitov K.V., Yanev G.P., Maximum individual score in critical two-type branching processes // C. R. Acad. Bulg. Sci., 2002, v. 55, 11, p. 1722. [6] Yanev G.P., Revisiting offspring maxima in branching processes // Pliska Studia Mathematica Bulgarica, 2007, v. 18, p. 401426. [7] Bertoin J., On the maximal offspring in a critical branching processes with infinite variance // J. Appl. Probab., 2011, v. 48, 2, p. 576582. 134

..

[8] Bertoin J., On the maximal offspring in a critical branching processes with finite variance // J. Appl. Probab., 2013, v. 50, 3, p. 791800. [9] Lebedev A.V., Maxima of random particles scores in Markov branching processes with continuous time // Extremes, 2008, v. 11, 2, p. 203216. [10] .., // . -. . 1. . . 2008, 5, . 36. [11] .., // , 2012, . 57, 4, . 788794. [12] .., // . . - , 2010, 1, . 714. http://www.math.msu.su/department/probab/linasl.pdf [13] O'Connel N., The genealogy of branching processes and the age of our most recent common ancestor // Adv. Appl. Probab., 1995, v. 27, p. 418442. [14] Adami C., Chu J., Critical and near-critical branching processes // Phys. Rev., 2002, v. 66, 011907. [15] Budhiraja A., Reinhold D., Near critical catalyst-reactant process with controlled immigration // Adv. Appl. Probab., 2013, v. 23, 5, p. 20532098. [16] Nelsen R., An introduction to copulas. Springer, 2006. [17] ., ., ., . .: , 1989. [18] ., . .: , 1985. [19] Fleischmann K., Siegmund-Schultze R., The structure of reduced critical Galton-Watson processes // Math. Nachr., 1977, v. 79, 1, p. 233241. [20] .., // , 1980, . 25, 3, . 593596. [21] .., // , 1985, . 30, 1, . 183188.

135

1
..2

( ), . , . . , .

1

[1] [2, 3] ( ) T R x f y = f (f -1 (x) + f -1 (y )), x, y T , (1) f : U T , U = R U = R+ . f (r + s) = f (r) f f (s), r, s U, . , z = x f y x y , .. " ": y = f (f -1 (z ) - f -1 (x)). , , : x > y x y x y , . f (u) = eu , , (1) T = (0, +). [4] (1) f (u) = -h ln u, u > 0, . [5] (1) f (u) = |u|s sign u, s > 0 . T R+ . (1) , [6, . 4].
14-01-00075. , avlebed@yandex.ru, , , . .. .
2 1

136

.. ...

, [6]. C [0, 1]d , d 2, [0, 1]. , Rd F (x1 , . . . xd ) = C (F1 (x1 ), . . . Fd (xd )), Fi , 1 i d . , . , . (d = 2). C (u, v ), u, v [0, 1], , : (1) C (u, 0) = C (0, v ) = 0; (C2) C (u, 1) = u, C (1, v ) = v ; (C3) C (u2 , v2 ) - C (u2 , v1 ) - C (u1 , v2 ) + C (u1 , v1 ) 0, u1 u2 , v1 v2 . , (C4) C (u1 , v1 ) C (u2 , v2 ), u1 u2 , v1 v2 , (C1)(C3). , -: (C5) max{u + v - 1, 0} C (u, v ) min{u, v }. , . (R, 1 - R) (R, R), R [0, 1]. ^ C X Y , , : ^ P(X > x, Y > y ) = C (P(X > x), P(Y > y )). ^ C C ^ C (u, v ) = u + v - 1 + C (1 - u, 1 - v ), . , C (u, v ) > 0 u, v (0, 1). , C (u, v ) = -1 ((u) + (v )), (2)

: [0, 1] R+ , , , (0) = +, (1) = 0, . , . . (0) < , [-1] : t (0) [-1] (t) = 0, . ( [0, 1]). , , , C (u, u) < u u (0, 1), . . 137

2

,

R+ , , : x C y = - ln C (e-x , e-y ), x, y 0. (3) . , , .. . , (1), f : f (s) = - ln -1 (s), (t) = f
-1

(- ln t).

(4)

(4) f : f R+ , f (0) = 0, f (+) = +. , (1) U = T = R+ . (4): (X, Y ) C , Z , P(Z > x C y ) = P(X > x, Y > y ). , (C1)(C5) (3): (O1) x C y +, x + y +; (O2) x C 0 = x, 0 C y = y ; (O3) exp{-x2 C y2 } - exp{-x2 C y1 } - exp{-x1 C y2 } + exp{-x1 C y1 } 0, x1 x2 , y1 y2 ; (O4) x1 C y1 x2 C y2 , x1 x2 , y1 y2 ; (O5) max{x, y } x C y - ln max{e-x + e-y - 1, 0}. e-x + e-y - 1 0 . . 2.1. - C (u, v ) = exp - (- ln u) + (- ln v ) (t) = (- ln u) , f (s) = s
1/ 1/

,

1,

,

x C y = (x + y )1/ . 2.2. C (u, v ) = u
-

+v

-

-1

1/

,

> 0,

(t) = t- - 1, f (s) = (1/) ln(1 + s), x C y = 1 ln(e
x

+ ey - 1).

2.3. - C (u, v ) = uv exp{- ln u ln v }, 138 0 1,

.. ...

(t) = ln(1 - ln t), f (s) = (es - 1)/, x C y = x + y + xy . 2.4. -- C (u, v ) = uv , 1 - (1 - u)(1 - v ) || 1,

(t) = ln((1 - (1 - t))/t), f (s) = ln((es - )/(1 - )), x C y = ln (1 - )ex 2.5. C (u, v ) = - ln 1 + (t) = - ln((e-t - 1)/(e
- +y

+ (ex + ey - 1) .

(e-

u

- 1)(e-v - 1) e- - 1

,

- 1)), f (s) = - ln(-(1/) ln(1 + e-s (e- - 1))),
- e-x

(e 1 x C y = - ln - ln 1 +

- 1)(e- e- - 1

e

-y

- 1)

.

, .. , (x) (y ) = (x y ), C (u , v ) = C (u, v ), [6, §3.3.4]. - ( 2.1), x y = (x + y )1/ . (free-scale), () . , C (u, v ) = uv x + y . , , , C . . , , C C [6, . 5]:
1 1 1 1

> 0.

(5)

C = 4
0 0

C (u, v ) dC (u, v ) - 1,

C = 12
0 0

C (u, v ) du dv - 3,

C = 4E exp{-X C Y } - 1, C = 12E exp{-Z C W } - 3,

(X, Y ) C , (Z, W ) . , 139

R+ , [-1, 1]. C (u, v ) uv C (u, v ) uv . x C y x + y x C y x + y . 2.1 2.2 , 2.3 , 2.4 2.5 > 0 < 0. , : , .

3

" " [3]: k , 1 k m, Xn Xn-1 = xk , Xn 1 · · · m Xn-1 = x1 + · · · + xm , xk 0, m 2. , - . , [7, 8], Xn = n (X
n-1

),

n (t), t 0, (t). ( , .), ( , ..). , , , [9] [10]. (1) [2, 3]. : Xn = f (n (X
n-1

)),

(6)

n (t), t 0, (t). (t) (6) V . (6) , : V U , T R+ . (6) T . Yn = f -1 (Xn ), Yn = n (f (Yn-1 )), (7) V , - [11]. ( , ..), , . , 140

.. ...

; "" , . , : (1) . , [3], A > 0 f (x) = A(1 - e-x/A ). , ; (O2) , , ; (O3) ; (O4) , , ; (5) ( ), , . , (.. ). . , . , x y x + y ( ), (. 2.1, 2.2 2.4, 2.5 > 0). "" , . [12]. , , "" . 3.1. (t), t 0 , c - Ee-t(1) = e-t , 0 < < 1, f (s) = s1/ , 1 (. 2.1). (6) f , : Xn = X
d 1/() 1/ n-1 n

,

n , n 1, n = (1). , > 1 ( ). 0, > 1 ( ) , Xn = X
1/ n-1 n

,

(8)

n , n 1, (x) = exp{-x- }, x > 0 ( - [12]). (8) [9] c , . , = 1 Xn = X
1/ n n

,

141

E ln (1). , , , [13]. . (6) , , . , , , , , , , , f . (7). , f (x) x , . f (x) > x , , , , ( ), .. .

4

[7, 8] Zn = n (Zn-1 ). (9)

, , , . , , , ( ). f () = - ln Ee-(1) , 0. ( ). [14, 2.2] x > 0: e-x , = sup{ 0 : f () = }, , = sup{ 0 : f () }, (1) 1 .. f () 0, = + .. P( (1) < 1) > 0 0 < , (9). (6) ( ). , (6): , . µs = E (1)s , s > 0. 1. N > 0, > 0, 0 < s 1, µs < f (x) xs /µs - x N , P(Xn +) = 0. 142

.. ...

. [15, . 1, §4.2, 1]. , E (t)s µs t, 0 < s 1, t 0, x N : E(Xn |Xn-1 = x) - x = E(f ( (x)) - x -, , [0, N ) x x/. , .. r = inf {u > 0 : P( (1) u) > 0}, (1) r (t) rt .. P( (1) < 1) > 0 r < 1. 1. f (x) x, r < 1 1, P(Xn 0) = 1. . , Xn Zn . , Xn Zn , n 1 .. P( (1) < 1) > 0, 0 < . = 0, Zn .., , Xn . > 0, Zn , . , V = [0, N ), , V . - Zn V , 1 . Xn V , 1 , , X1 V , 1 = 1 . Zn , 1 , X1 , , (2) (2) (2) Zn , Xn Zn-1 , n 1 . , Zn V ( 2 ..), ( Xn ). ( ), Xn .. 4.1. [3] , f (x) = 1 - e-x , (1) . 1 .. [3] ( ). , . 2. > 0, f (rx) x + v (x) x (0, ), v ( x) , v (x) > 0, x > 0. P(Xn 0) = 0. . X0 = x0 > 0. xn = f (rxn-1 ), n 1, Xn xn , n 1 .. x0 < , n , xn-1 < , xn x0 + nv (x0 ), xn (0, ) ( - x0 )/v (x0 ) , , Xn .. 4.2. f (x) = 1 - e-x , 2 r > 1, r < 1 1. r = 1 . 4.3. f (x) = 1 - exp{-x - x2 }, > 1/2, 2 r 1. 143

4.4. f (x) = x1/ , > 1, 2 r > 0. 4.2 4.3 (). 4.4 1 , s > 1/ , µs < , 0 < µ1/ < 1. 2. 2 N > 0 , f (rN ) < N , x = inf {x > 0 : f (rx) = x} [, N ) P(lim inf Xn x ) = 1.
n

. 0 < x < f (rx) > x, f (rN ) < N , - [, N ) , f (rx) = x. x , x < f (rx) < x x < x f (rx) x x x . xn , 2, xn x x0 < x xn x x0 x . lim inf xn x , lim inf Xn x ..
n n

, 1 2 , , .. , . (6) [2]. , (1) . [15, 8, . 1, §2], . , . 3. X0 = x > 0. : 1) f (x) x, P(Xn 0) e 2) f (x) x, P(Xn 0) e
- x - x

, P(Xn ) 1 - e-x ; , P(Xn ) 1 - e-x .

. Xn Zn , Xn Zn , n 1 .. Xn Zn , n 1 .. .

5

. , a b, n, na > b. : u, v (0, 1) un = C (u, un-1 ), u1 = u, n, un < v . , "", , , . , , . (1). a, b T , f -1 (a), f -1 (b) > 0, an = a f an-1 , a1 = a, an = f (nf -1 (a)). , 144

.. ...

f , an b, f , , b, n. . [2] . f : R T , (-, 0) (0, +), T R+ . f -1 (t) , f (s) = t, , t = f (0). (1), f -1 (x) f -1 (y ) , , ± . , f . [1, §1.4] , f (s) = |s|1/ , > 0. , . , , , , , "" ( ) . ( ) , [16]. g : R+ T g (s) = f (s) s 0, f (s) = g (|s|) : x ± y = g (|1 g -1 (x) + 2 g -1 (y )|), f (10)

i , i 1, , ±1. a T , g -1 (a) > 0, an = a ± an-1 , f a1 = a,
n

an = g g (a)
i=1

-1

i

,

, an g () n . b T , g -1 (b) > 0, > 0 n, P(an > b) > 1 - ( g ) , P(an < b) > 1 - ( g ). ± , f , an > b an < b ( ). [2] ± . f , (t) f (s) = + s2 , , > 0, , Yn , n 1, ARCH- , [17, . 2, §3a]: Yn = Xn , n 1, + Y
2 n-1 n

,

: Xn = + (2 )X n 145
n-1

,

, - , . , {0, 1, . . . N }. N . k . k , CN . , . , , [18] [19]. k1 k2 k1 k2 . : k1 k2 = k2 k1 ,
d

(k1 k2 ) k3 = k1 (k2 k3 ).

d

: 1. k 0 = k . 2. k N = N (.. N " "). 3. k1 l1 k2 l2 , k1 k2 d l1 l2 ( ). 4. max(k1 , k2 ) k1 k2 k1 + k2 .. k , l {1, . . . , N - 1}, u1 = k , un = un-1 k , .. n( ), un = N , , un > l. , . :
Xn
-1

Xn =
i=1

i,n ,

0 X0 N ,

i,n , i, n 1, ... {0, 1, . . . N }. , , -. . , . : 1. P( = 0) = p0 > 0, P(Xn = 0|Xn-1 > 0) pN , 0 0 .. 2. P( = 1) = 1, X0 1 1 .. , [20], . 3. P( = 0) = 0 P( = 1) < 1, Xn .

[1] .., . .: , 1986. [2] .., // . -. . 1. . , 2005, 4, c. 35. [3] .., // . -. .1. . . 2006, 4, . 5658. 146

.. ...

[4] .., // , 2005, . 78, 3, . 377395. [5] .., // , 1979, . 96, . 7081. [6] Nelsen R., An introduction to copulas. Springer, 2006. [7] Ji rina M., Stochastic branching processes with continuous state space // Chechoslovak Math. J., 1958, v. 8, 2, p. 292313. [8] ., // , 1959, . 4, 4, . 482484. [9] .., // , 2005, .50, 3, . 564570. [10] .. // , 2009, . 4, 1, . 93106. http://www.math.msu.su/department/probab/svodny2.pdf [11] .., .., // , 1974, . 19, 1, . 1524. [12] .., - - // , 2013, . 8, 3, . 102107. http://www.math.msu.su/department/probab/mm80.pdf [13] .., // , 2002, . 57, 2, . 2384. [14] Lambert A., The contour of splitting trees is a Levy process // Ann. Probab., 2010, v. 38, 1, p. 348395. [15] .. . .: , 1999. [16] ., // , 1985, v. 40, 4, p. 205206. [17] .., . .1. . . .: , 1998. [18] .., .., .., C . .: , 1976. [19] .., .., // , 1982, . 27, 4, c. 684692. [20] .., .., // , 2009, 3, c. 373381.

147

..1 , ..2 , ..
3

( ) , .

1

, , , . , () ( ), , . , , , : 1. () . , . , , (., [5, 1, 4, 13, 2, 14, 3]). , , . .., .. .. , , , [7, 8, 9, 6]. 2. , [12], [17, 18, 16]. 3. , ( ), [11], [5].
, alekslyk@yandex.ru, , , - . .. . 2 , 2malyshev@mail.ru, , (. .) , - . .. . 3 , magaarm@list.ru, , . .. .
1

148

.., .., ..

, , , , . , ( ). , . , , ( ). , , . (. [10]) . , , , , d , . .

2

, , , , . . t 0 , , , : ... < zN (t) < ... < z1 (t) < z0 (t). , , 0, , v0 (t), . . " ", , , k = 1, 2..., k - 1. , , . . -, k rk (t) = zk-1 (t) - zk (t) , k - 1, d > 0 (, ). , d Fk d. Fk (t) = 2 (rk (t) - d) = 2 (zk-1 (t) - zk (t) - d), k - , rk (t) , , . -, -vk (t) . , . 149

. -, = 0, . , , , . , , . zk (t) = 2 (zk-1 (t) - zk (t) - d) - zk (t), k = 1, 2, ... (1)

v0 (t), a 2 , . : zk (0) = -k d, zk (0) = v

v 0. xk (t) = zk-1 (t) - zk (t) - d, , zk (t) = 2 xk (t) - zk (t) = 2 xk (t) - (-xk (t) + zk-1 (t)), zk (t) = -xk (t) + zk-1 (t), (1) xk (t) + xk (t) + 2 xk (t) = 2 x xk (0) = 0, x0 (t) = 1 (z (t) + z0 (t)). 2 0
k-1

(t), k = 1, 2, ...

(2)

xk (0) = 0

3

1. |x0 (t)| c t 0 2 > 0. k t 0 : |xk (t)| c . - Xk (t) = (xk (t), xk (t))T . (2) : Xk (t) = AXk (t) + 2 x e2 = (0, 1)
T k -1

(t)e2 ,

, A : A= 0 -
2

1 -

.

t

Xk (t) =

2 0

x

k -1

(s) exp(A(t - s))e2 ds.

(3)

150

.., .., ..

A. . 1. 2 > 4 2 . Q() = 2 + + 2 = 0 , 1 , 2 : 1 < 2 < 0, (4)

1 1 , 1 2 . : = exp(At) = (t) -1 (0) = (exp(At)e2 )1 = exp(1 t) exp(2 t) 1 exp(1 t) 2 exp(2 t) 2 - -1 1 1 2 - exp(1 t) exp(2 t) 1 exp(1 t) 2 exp(2 t) ,

1

1

1 (exp(2 t) - exp(1 t)), 2 - 1

(exp(At)e2 )1 . : t 2 xk (t) = (exp(2 (t - s)) - exp(1 (t - s)))xk-1 (s)ds. (5) 2 - 1 0 k . , k - 1, k . (3), 1 2 = 2 , : 2 |xk (t)| = | 2 - 1 c 2 2 -
+ t

(exp(2 s) - exp(1 s))x
0

k-1

(t - s)ds|

(exp(2 s) - exp(1 s))ds =
1 0

c 2 1 1 ( - ) = c. 2 - 1 1 2

2. = 2 . : 1 = 2 = - . 2 1 ( ): (exp(At)e2 )1 =
t

1 (exp(2 t) - exp(1 t)) - t exp(- t). 2 - 1
2 0

xk (t) =

(t - s) exp(- (t - s))x

k -1

(s)ds

t

|xk (t)| = |
0

2

(t - s) exp(- (t - s))x

k -1

(s)ds|

c 2 | exp(- t)(1 + t) - 1| c. 2

: xk (0) = ak , k N, xk (0) = bk , k N, |ak | a, |bk | b k a, b. 151

2. |x0 (t)| c t 0 , > 2 , = k t 0 : |xk (t)| max{c, C }.

2 4

- 2, C =

a+2b 2

.

. . , 2 ± = - ± - 2 2 4 (4) , xk (t) = x x
k,± k,+

(t) + x

k,-

(t),

(t) = C

k,±

e

± t

e± t + Q (± )
2 2

t

e-
0 t

± t1

x

k-1

(t1 )dt1 =

=C

k,±

e

±

t

+

e ± t 2± +

e-
0

± t1

x

k-1

(t1 )dt1 .

, C
k,±

=

1 2

±

+ ak ± b 2

k

.

+ > - : |xk (t)| |C
k,+

| exp(+ t)+|C
0

k,-

| exp(- t)+sup |x
s0

k-1

(s)|

2 2

t

| exp(+ (t-t1 ))-exp(- (t-t1 ))|dt1
0

yj = sups

|xj (s)| yk . |C
k,±

|

( ± )a + b 2 , 2

, : t - t1 = s, : |xk (t)| |C 1 2
k,+

| exp(+ t) + |C

k,-

| exp(- t) + sup |x
s0

k -1

(s)|

2 2

t

(exp(+ s) - exp(- s))ds
0

1 - a + b + yk-1 - exp(+ t) + 2 2

+ a+b-y 2

k - 1 +

exp(- t) + y

k -1

.

. 1. f (t) = a exp(+ t) + b exp(- t) + c b, c > 0, a R, - < + < 0. t 0 : |f (t)| max{c, a + b + c}. 152

.., .., ..

. . 1. a > 0. , sups0 |f (s)| 0 a + b + c. 2. a < 0. t, 0, : t> a 1 ln - - - + b
+ -

+ 1 t0 = - -+ ln(- a - ) - . t0 < 0, sups0 |f (s)| b + c. t0 > 0, sups0 |f (s)| 0, +.

xk , yk max y .
k-1

,

a + 2b 2

,

4

xk (0) = xk (0) = 0, k = 1, 2, . . .

(2)

> 0, > 0, x0 (t). , k t , , t = µk , k µ > 0. 3. x0 (t) = v t 0. > 0 > 0 q± > 1, µ± > 0, c± = 0, , c± xk (t) q k
±k ±

, t = µ± k , k .

d > 0 , , . . :

uk (z ) =
0

e

-z t

xk (t)dt,

Re(z ) > 0, k = 0, 1, . . . .

(2) : uk (z ) = uk (z ) = 2 z 2 + z +
2

2 uk-1 (z ), k = 1, 2, . . . . z 2 + z + 2
k

u0 (z ).

, < 2 . - . 2 153

uk (a + ib) b R1 a > - . , 2 k t > 0 : xk (t) = 1 2 i
a+i

uk (z )ezt dz ,
a-i

a > - . t = µk : 2 v xk (µk ) = F (µ, k ) = 2 i = v 2 i
a+i a-i a+i a-i

1 exp k (µz - ln(z 2 + z + 2 ) + ln 2 ) dz = z S (z ) = µz - ln(z 2 + z + 2 ) + ln 2 . (6)

1 exp(k S (z ))dz , z

z 2 + z + 2 a + ib, b R . , a > - 2 , : ln(z 2 + z + 2 ) = ln |z 2 + z + 2 | + i arg(z 2 + z + 2 ). 2z + =0 z 2 + z + 2 S (z ) = µ - µz 2 + (µ - 2)z + 2 µ - = 0, D = (µ - 2)2 - 4µ( 2 µ - ) = 2 µ2 + 4 - 4µ2 2 = -4µ2 r2 + 4, r2 = 2 - 2 > 0. 4 (7)

, µ < 1 . D > 0 ( r , , , ): 2 - µ + D 1 1 z = z (µ) = =- + + 1 - µ2 r 2 . 2µ 2µµ , z (µ) . S (z ): S (z (µ)) = - =- 2 (2z + )2 +2 |z z 2 + z + 2 (z + z + 2 )2
2 =z (µ)

=- .

2µ + µ2 = 2z (µ) +

µ
1 µ

+

1 µ

1 - µ2 r

+ µ2 = µ

2

1 - µ2 r 1+

2 2

1 - µ2 r

1 , µ < d , , z (µ) . (6) a = z (µ).

F (µ, k ) =

v 2 i

z (µ)+i z (µ)-i

1 exp(k S (z ))dz . z

154

.., .., ..

, z (µ) + ib, b R S (z ). : 1) : max Re(S (z (µ) + iy )) = Re(S (z (µ));
y R

(8)

2) z (µ) , Re(S (z )) < Re(S (z (µ)). . : Re(S (z (µ) + iy )) = µz (µ) - = µz (µ) + ln 2 - 1 ln (z 2 (µ) - y 2 + z (µ) + 2 )2 + (2z (µ)y + y )2 + ln 2 = 2 h(s) = (z 2 (µ) - s + z (µ) + 2 )2 + s(2z (µ) + )2 .

1 ln h(y 2 ), 2

h(s) : h(s) = s2 + s (2z (µ) + )2 - 2(z 2 (µ) + z (µ) + 2 ) + (z 2 (µ) + z (µ) + 2 )2 . h(s) : (2z (µ) + )2 - 2(z 2 (µ) + z (µ s0 = - 2 1 1 = -(2z (µ) + )( + 1 - µ2 r 2 - µµ , s0 < 0. h(s) s s = 0. , : 0 ) + 2) =- (2z (µ) + )2 - 2 2 1 - µ2 r 2 ) 1 µ
(2z (µ)+) µ

=

2 2 1 ) = -( + µ µµ

1 - µ2 r 2 .

1 1 max Re(S (z (µ) + iy )) = µz (µ) + ln 2 - ln min h(s) = µz (µ) + ln 2 - ln h(0) = Re(S (z (µ)). y R 2 s0 2 , (8) . , z (µ) f (a) = Re(S (a)) a > 0. : f (a) = Re(S (a)) = µa - ln a2 + a + : f (a) = µ - µ(a - z (µ))(a - z (µ)) ~ 2a + = , 2 2 + a + 2 a + a + a
2 2

+ ln 2 .

z (µ) (7), , z (µ) < ~ ~ z (µ). , f (z (µ), z (µ)) ~ (z (µ), +), , , z (µ) f (a). . , 1.3 . 263 [15] : F (µ, k ) cekS (z(µ)) , k , 155

c : c= 1 v . 2 k S (z (µ)) z (µ)
1 d

, > 0 µ < S (z (µ)) = - =- µ =
1 r

S (z (µ)) . : 2z (µ) + µ + ln 2 =

µ +1+ 2

1 - µ2 r2 - ln

µ +1+ 2

1 - µ2 r2 - ln(1 +

1 - µ2 r2 ) + ln(µ2 2 ).

: Sz 1 r =1- + ln 2r 2 r2 1, .

, S (z ( 1 )) > 0. , r S (z (µ)) , µ = µ+ , S (z (µ)) . µ- , S (z (µ)) - µ 0. .

[1] Haight F., Mathematical theories of traffic flow. Elsevier, 1963. [2] Renyi A, On two mathematical models of the traffic on a divided highway // Journal of Applied Probability, 1964, v. 1, p. 311320. [3] Solomon H., Wang P., Nonhomogeneous Poisson fields of random lines with applications to traffic flow // Proc. Sixth Berkeley Symp. on Math. Statist. and Prob. (Univ. of Calif. Press), 1972, v. 3, p. 383400. [4] Kel ly F., Reversibility and stochastic networks. N.-Y.: Wiley. 1979. [5] Caceres F., Ferrari P., Pechersky E., A slow-to-start traffic model related to a M/M/1 queue // Journal of Statistical Mechanics: Theory and Experiment. 2007, P07008. Available at http://iopscience.iop.org/1742-5468/2007/07/P07008/fulltext/ [6] .., .., .., : . . 2, . . , 2013. [7] .., .., . 1966. [8] / . - . , 1975. [9] / 2 - . , 1970. 156

.., .., ..

[10] .., .., ., . " " . http://www.genplanmos.ru. [11] Blank M., Ergodic properties of a simple deterministic traffic flow model // J. Stat. Phys., 2003, v. 111, p. 903930. [12] Lighthil l M.J., Whitham G.B., On kinematic waves. II // Theory of traffic flow on long crowded roads. Proc. R. Soc. London, Se. A., 1955, v. 229, p. 281345. [13] Malyshev V., Yakovlev A., Condensation in large closed Jackson networks // Ann. Appl. Prob., 1996, v. 6, 1, p. 92115. [14] Serfozo R., Introduction to stochastic networks. Springer, 1999. [15] .. . . .: , 1987. [16] Prigogine I., Herman R., Kinetic theory of vehicular traffic. N.Y.: Elsevier, 1971. [17] Helbing D., Verkehrsdynamik. Berlin: Springer, 1997. [18] Helbing D., Traffic and related self-driven many particle systems // Rev. Mod. Phys., 2001, v. 73, p. 10671141.

157

. . .-.
..
1

. . , 1937 ., . , .-. 1956 . . . , . , , , , .

X1 , X2 , . . . - , P . Pn ···+ X1 +n Xn , 2 (x) = e-x /2 / 2 . () , n (Fn , G) = sup{|Fn (x) - G(x)| : - < x < } 0, Fn (x) G(x) Pn . , , , , . . 2 1937 . [12] , P . , . . , P p - pq = q, P q pq = p, q = 1 - p, 0 < p < 1.

, P , supp P = {x : P ((x - , x + )) > 0 > 0} ( ) D = {a + k h : k = 0, ±1, ±2, . . . }
, v.senatov@yandex.ru, , , - . .. . 2 (1883 1947) . , . (1921). 1930 . , .
1

158

.. . . .-.

h > 0. D a D. P , h , P , . , P h, Pn h/ n, Dn = {x : an + k h/ n, k = 0, ±1, ±2, . . . }, an = a n, Fn (x) Dn , . p h = 1/ pq .

k =
-

xk P (dx),

k = 1, 2, . . . ,

s =

|x|s P (dx),
-

s>0

P . p q-p 3 = , pq p2 + q 2 3 = , pq 4 = 4 = p3 + q 3 1 = - 3. pq pq

. . (. [5, c. 223] [6, c. 95]) , P p npq 25. 1 3 Fn (x) - G(x) = (1 - x2 )e 6 2 n ¯ Dn = Dn +
h 2n -
x2 2

+ OI

1 n

,

¯ x Dn ,

(1)

Dn , OI 1 n 0.13 + 0.18 |p - q | +e npq
-3 npq /2

.

(2)

. . (. [1]) 1 3 pn (x) - (x) = (x3 - 3x)e ~ 6 2 n
-x2 /2

+ OL

1 n

,

x Dn ,

(3)

n p n ( x) = ~ (Fn (x + 0) - Fn (x - 0)), h 1 0.15 + 0.25 |p - q | OL +e n npq

x Dn ,
-3 npq /2

.

(4)

. n , (3).

I , , L , (1) (3) , . , 3 = 0 x ±1 (1) 0 ± 3

159

, (1) (3) P . (2) (4), . (2) (4), , , . , . 1937 . . [3], , P limt |f (t)| < 1, f (t) P , 1 3 1 2 Fn (x) - G(x) = (1 - x2 )e-x /2 + o , n , (5) n 6 2 n
1 o( n ) x. (5) , 3 = 0 x = ±1 . , P , P . . , . . [4], ,

3 (Fn , G) c , n c > 0 . 1941 . . 1942 . .-. , P . . . . (1) . (5) .-. 1945 . . , P . , , . , (5) h Sn (x)(x), 2n Sn (x) , Dn -1 1, Dn 1 -1. ¯ G(x) Fn (x). Dn Sn (x) = 0. 1 o( n ) .-. . c . .-. , 1956 . , 3 + 10 c cE = = 0.4097 . . . . 6 2
1 , cE 62 , (1). cE .

160

.. . . .-.

, , p = (4 - 10)/2, . , P , ( , ) n 3 (Fn , G) cE . n , , .-. . 2001 . . . [9] 3 (Fn , G) cE + C n 3 n
40/39

3 log( ) n

7/6

,

C ( . . , ). 2012 . . . [11] , 3 (Fn , G) cE + 2.58 n 3 n
2

.

, , , Fn (x) G(x). , (1) (2) , , , , , . o(1/ n) (5), n O(1/n) P , [7] [8]. , ( ), , . , , , [5, . 232]. ¯ (1) (2) , p x Dn 1 |3 | |Fn (x) - G(x)| |1 - x2 |e- 6 2 n
x2 /2

+

0.13 + 0.18|p - q | +e npq

-3 npq /2

.

(6)

p = 0.4188 . . . , h 0.022 0.655 max{|Fn (x) - G(x)| : x Dn + } + + e- n 2n n 161
0.74 n

(7)

( 2 |1 - x2 |e-x /2 , |x| 0.63 0.5, - ). , 3 > 1.04 1.04 0.426 3 cE > 0.4097 = . n n n (7) 19. , - : , Fn (x) G(x), Fn (x) Fn (x) G(x) . , Fn (x) x, Fn , x, Fn (x) . x Dn , . c , P , , 1956 . c (1). , p P (0, 0.5). (xn , xn ), xn , xn Dn , Fn , x = n 1 (xn + xn )/2 . xn , x , xn O( n ). (1) , n 1 q-p +o >0 Fn (x ) - G(x ) = n n n 6 2 npq n. xn , Fn (x) h 1 , G(x) 2n G (0) + o n . h q-p 1 1 + +o Fn (xn + 0) - G(xn + 0) = 6 npq 2 2 n 2 1 q-p+3 = +o c n 6 2 npq c . q-p+3 = cp c. 63 2 pq 3 =
p2 +q pq
2

1 n

=

3 , n ,

, ,

q-p+3 2-p cp = = . 6 2 (p2 + q 2 ) 3 2 (p2 + (1 - p)2 ) c p. p = 4-2 10 , . . . cE . q-p 1 h 1 + +o Fn (xn + 0) - G(xn + 0) = 6 npq 2 2 n 2 162 1 n =

.. . . .-.

1 h 1 |3 | = + +o 6 2 n 2 n 2

1 n

.

( ), (Fn , G) , P : (Fn , G) ¯ max{|Fn (x) - G(x)| : x Dn } Fn (. [2, . 3, §5]). 18.47 . , Fn (x), , . Fn (x) , , . Ga,h (x), , n Fn (x), G(x) . , (6) 1 |3 | 0.13 + 0.18|p - q | +e (Fn , Ga,h ) + n npq 6 2 n
-3 npq /2

.

, n D Ga,h , Fn (x), G(x), , n P , Fn (x), , . , : , , , n. . , P , , (1) . , , , . . Bk
2

,n,j

1 = 2

T

n

-T

|t|k µ
n

n-j

t n

dt,
2

t µ(t) = max{|f (t)|, et /2 }, k > 0, 0 j < n. , µn-j n e-t /2 n j t. , T , n

Bk

,n,j

Bk =

1 2

|t|k e-
-

t2 /2

dt.

(8)

k . 163

1. P , , h n 2 1 3 Fn (x) - G(x) = (1 - x2 )e 6 2 n OI 1 n 1 6 h 2n
5/2 2 -x2 /2

+O

I

1 n

,

¯ x Dn ,

4 + 3 1 B1 + B3,n,1 + 2 4!n 2 2 h 2n
2

3 3! n

2

B5,n,2 + 2

|3 | B4 + 12n3/2 2

n n-1 +

1 |3 | + 6 3! n
-
2 n 2 h2

1 |3 | 4 + 3 B4 + B6,n,2 + 2 2 3! n 4!n 2 t2 e
-t2 /2

h2 e 3n

+

|3 | 1 3! n

n h

dt,

(9)

, 3 pn (x) - (x) = (x3 - 3x)e- ~ 6 2 n OL + 1 n 4 + 3 1 B4,n,1 + 4!n 2 n n-1
2 n/2h2 3 x2 /2

+ OL
2

1 n

,

x Dn ,

3 3! n

B6,n,2 +

|3 | B5 12n3/2 h- e 2n

+

1 |3 | 4 + 3 B7,n,2 + 2 3! n 4!n

n h

+

+

|3 | 1 3! n

t3 e-

t2 /2

dt,

(10)

Bk

,n,j

T = . h

, (9) , (10) , , n . , . , . . . 1 , Bk,n,j . . , P h, |f (t)| e- µ(t) = e 0 j < n Bk
,n,j t2 /2

|t| , h |t|
h

-t2 /2

1 < 2

-

|t| e

k -(n-j )t2 /(2n)

dt =

n n-j

(k+1)/2

Bk ,

164

.. . . .-.
l- B2l = (221)!! , l = 1, 2, . . . , B2l+1 = 2l! , l = 1, 2, . . . . n 3
l

OL

1 n

1 1 8 2 npq + 2 |p - q | 3 n npq 1 e- 2 npq

n n-1 n n-2
3

5/2

5 (p - q ) + 24 2 npq

2

n n-1
4

7/2

+

+ +

1 |p - q | 6 (npq )3/2
npq

n n-2 t3 e
-t2 /2

+ dt.

+

2 npq /2

|p - q | 1 3! npq

|f (t)| , . , , . , 1 n , , , . OL 1 n 1 1 5 (p - q )2 1 |p - q |(1 + 4pq ) + + + 6 (npq )3/2 8 2 npq 24 2 npq 1 e- + 2 npq
2 npq /2

|p - q | 1 + 3! npq

npq

t3 e

-t2 /2

dt.

1 5 82 < 0.05, 242 < 0.084, 61 < 0.054, , , (4). 1 , ,

OI

1 n +

11 1 (p - q )2 |p - q |(1 + 1.2pq ) + + + 16 npq 18 npq 96 (npq )3/2 h2 e 3n
- 2 n/2h2

+

|3 | 1 3! n

n h

t2 e

-t2 /2

dt,

, (2). , . , 2 g (t) e-t /2 . t , g (t) = g n n n. = (t) ( , ), , , , , = , + = , . 2 h(t) = H (t) |h(t)| |H (t)|. 1. P , t f n n - g (t) n 2 f
n

t n

3 - g (t) = 3!

it n

3

ng

n-1

t n 165

4 + 3 + 4!

t n

4

nµn-

1

t n

+

+

3 3!

2

t n

6 2 Cn µ n-2

t n

+

3 4 + 3 3! 4!

t n

7 2 Cn µ n-2

t n

.

(11)

k-1

a -b =
j =0

k

k

a

k-j -1 j

b (a - b),

(12)

a, b k . f (t) g (t) , f (t) = 1 - t2 3 4 + (it)3 + t4 , 2 3! 4! g (t) = 1 - t2 t4 + , 2 8 - < t < .

, f
n-1 n

t n

- g (t) = f t n t n

n

t n f

-g t n

n

t n -g

= t n

=
j =0

f

n-j -1

g

j

t n g
j

= t n

=

it n

3

3 3!

n-1

f
j =0

n-j -1

t n

+

4 + 3 4!

t n

4

nµ

n-1

.

j , j = n - 1, it g n-1 n . , 0 j n - 2 (12) , f
n-j -1

it n

=f

n-j -1

it n t n

-g

n-j -1

it n -g
3

+g t n

n-j -1

it n
n-j -1

= t n

n-j -2

=
l=0

f

n-j -1-l-1

t n t n f

g

l

f

t n it n

+g

,

f 3 3! it n
3 n

3 - g (t) = 3! t n g

ng t n
-1

n-1

t n t n ,

+ t n

n-2 n-j1 -j2 -2

j1 +j2

f t n

-g

+

j1 +j2 =0

4 + 3 + 4!

t n

4

nµn

j1 , j2 0 j1 + j2 n - 2 2 , Cn . , n 2 f
n

t n

3 - g (t) = 3!

it n

3

ng

n-1

t n 166

4 + 3 + 4!

t n

4

nµ

n-1

t n

+

.. . . .-.

+

3 3!

2

t n

6 2 Cn µ n-2

t n

+

3 4 + 3 3! 4!

t n

7 2 Cn µ n-2

t n

,

. i h h -itx f (t) - g (t) F (x) - G(x) = e dt+ 2 2 - sin th 2 h ih + 2 2
h

e-itx g (t)

-

h

1 sin

th 2

-

1
th 2

dt +

1 2

e
|t|>
h

-itx

g (t) dt, it

x D + h , 2 h > 0, D = {a + k h : k = 0, ±1, ±2, . . . }, G(x) . [6, . 5, §2]. ¯ , n x Dn ih Fn (x) - G(x) = 2 2 n ih + 2 2 n (
n)/h -itx n/h - n/h

e

-itx

f

n

t n

- g (t)
th 2n

sin

dt+

-( n)/h

e +

g (t)

1 1 - sin(th/2 n) th/2 n e
-itx

dt+ (13)

1 2

|t|> n/h

g (t) dt. it

, 1 2 e
|t|> n/h -t2 /2

1 dt = |t|

n/h

te

-t2 /2

t2

dt

h2 3n

n/h

te

-t2 /2

dt =

h2 e 3n

-

2 n 2 h2

n . (13). , 1 1 th/2 n - sin(th/2 n) - = = sin(th/2 n) th/2 n sin(th/2 n)(th/2 n) th th/2 n (th/2 n)3 = ·· . = 6 2 n sin(th/2 n) 6 sin(th/2 n)(th/2 n) th/2 n sin(th/2 n)

(14)

(15)

t, t n/h 1 /2. 1h 2 2 n
n/h - n/h

e-

t2 /2

1 |t|h 1 dt < 62 n2 6

h 2n 167

2

1 2 2

|t|e-
-

t2 /2

dt =

1 h 2 B1 . 62 n 2

(13). t f n n -g (t), 1. , (11) , 1h 2 2 n
n/h - n/h

1 | sin (th/2 n) |
n/h - n/h

4 + 3 4 tµ 4!n

n-1

t n

+

1 2

3 3! n
n-2

2

t6 µ t n |t|5 µ

n-2

t n

dt+

1h + 2 2 n 1 = 2
n/h - n/h

1 1 |3 | 4 + 3 7 |t| µ | sin (th/2 n) | 2 3! n 4!n 4 + 3 3 n-1 1 t |t| µ + 4!n 2 n 1 |3 | 4 + 3 6 th2 n t sin (th/2 n) 2 3! n 4!n

dt = t n

th/2 n sin th/2 n 1 + 2
n/h - n/h

3 3! n µ
n-2

2 n-2

dt+

t n

dt.

, 4 + 3 1 B3,n,1 + 4!n 2 2 3 3! n
2

1 |3 | 4 + 3 B5,n,2 + B6,n,2 . 2 2 3! n 4!n 2

(11), . , 3 2 2 (it)3 e-t /2 et 3! n 3 (it)3 e- 3! n t2 e 2n 3 (it)3 e- 3! n 3 5 tg 12n3/2 t n

/2n

=

t2 /2

1+

t2 /2n

=

t2 /2

+

n-1

.

, n/h th/2 n |3 | 4 -(n-1)t2 /2n 1 te dt < 2 -n/h sin (th/2 n) 12n3/2 |3 | < 2 12n3/2 n n-1
5/2

1 2

-

te

4 -t2 /2

|3 | dt = B4 12n3/2 2

n n-1

5/2

.

, , ih 2 2 n
n/h - n/h

e-itx

1 3 (it)3 e- sin (th/2 n) 3! n

t2 /2

dt.

, ih 2 2 n
n/h - n/h

e-

itx

1 1 - sin (th/2 n) th/2 n
n/h - n/h

3 (it)3 e 3! n
-t2 /2

-t2 /2

dt+

ih + 2 2 n

e-

itx

1 3 (it)3 e th/2 n 3! n 168

dt.

(16)

.. . . .-.

(14), , 1 6 h 2n
2

1 2

n/h - n/h

th/2 n |3 | 4 - 2 1 |3 | t e t /2dt < 6 3! n sin (th/2 n) 3! n

h 2n

2

B4 . 2

(16) 1 - 2
n/h - n/h

e

-itx

3 (it)2 e- 3! n 1 2

t2 /2

1 dt = - 2
-itx

e-
-

itx

3 (it)2 e 3! n

-t2 /2

dt+

+

e

|t|> n/h

3 (it)2 e- 3! n

t2 /2

dt.

(17)

|3 | 1 3! n
n/h

t2 e

-t2 /2

dt,

n . (17) 1 2 3 (1 - x2 )e-x /2 . 6 2 n , , 1 2

e-itx itk e-
-

t2 /2

dt = Hk (x)(x), k = 0, 1, . . .

Hk (x) , , Hk (x) = (-1)k (k) (x)/(x), , H2 (x) = x2 - 1. . , x D h > 0, D = {a + k h : k = 0, ±1, ±2, . . . }, /h h e-itx f (t)dt. 2 -/h , n 1 pn (x) = ~ (Fn (x + 0) - Fn (x - 0)) = h 2 1 pn (x) - (x) = ~ 2
n/h - n/h n/h - n/h

e-itx f

n

t n

dt,

x Dn ,

e

-itx

f

n

t n

- g (t) dt -

1 2

|t|> n/h

e

-itx

g (t)dt,

x Dn .

, 169

1

n/h

e

-t2 /2

1 dt =

n/h

t- e t

t2 /2

h dt 2 n

n/h

te

-t2 /2

dt =

h- e n
2

2 n/2h2

.

, , 1. , (11) , 1 4 + 3 B4,n,1 + 4!n 2 3 3! n
2

B6,n,2 +

1 |3 | 4 + 3 B7,n,2 , 2 3! n 4!n

3!3n H3 (x)(x) , 3 3 n |3 | 2 t3 e-t /2 dt. B5 + 3/2 12n n-1 3! n n/h

, H3 (x) = x3 - 3x. , . , Bk,n,j |f (t)|. . , P p, |f (t)| = 1 - 4pq sin
t 2 pq 2

, 0.35 p 0.65 |t| /h = pq 2 2 |f (t)| e-t /2 . µ(t) = e-t /2 |t| /h p , . , 2 , , 0.25, |f (t)| e-t /2 |t| /h . 22 , 0 < 2 < 1 , |f (t)| e- t /2 |t| /h. , , , , Bk
-(k+1) ,n,j

-(k+1)

Bk

n n-j

(k+1)/2

,

. 0.25 2 0.75, Bk,n,j , 1.15k+1 , k 4 , . -. T |f (t)|. , 0.25 2 |f (t)| e-t /2 |t| T = 0.92. P h 2 |f (t)| e-t /2 |t| T , 0 < T /h. pn (x) - (x) ~ 1 pn (x) - (x) = ~ 2 + 1 2 e
-itx n T n

-T

e
n

-itx

t f n - g (t) dt+ n e-itx e-
n t2 /2

T

n<|t| n/h

f

t n

dt -

1 2

|t|>T

dt,

x Dn .

(18)

, 1 2 e-T n/2 , T n 170

.. . . .-.

n . (18) n (T ) = max{|f (t)| : T t /h} < 1, n . . , (T ) = 1, T > 0, , |f ( )| = 1 T /h. , P D h = 2 2h, , h D , P. (18) , Bk,n,j T , (18). , 2 P h |f (t)| e-t /2 |t| T , 0 < T /h, (10) OL 4 + 3 1 1 |3 | 4 + 3 3 |3 | 1 B4 + B+ B7 + B6 + 3/2 5 n 4!n 2 3! n 12n 2 3! n 4!n 1 nn |3 | 1 3 -t2 /2 2 e-T n/2 + + (T ) -T + te dt. h T n 3! n T n
1/4 2 /h

|f (t)| dt
n

T

nn (T ) -T , h

(19)

12 T n T = 4 +3 n-1/4 . , |f (t)|2 P P - , P - (A) = P (-A) A. P P - ( P ) , , , 2(4 + 3) , |f (t)|2 = 1 - t2 + 4 +3 t4 , 12 |f (t/ n)|2 1 - t2 /n + 4 +3 t4 /n2 , t2 n. 12 ,

f

t n

2

exp -t2 /n +

4 + 3 t 4 12 n2

,

(20)

4 + 3 t2 1. 12 n , |t| T n = 4 + 3 12n 12 4 + 3
1/2

12 4 +3

1/4

n

1/4

n

n

1/2

1.

+3 n 4, 4n 12. , , .

171

t µ t n
n-j

exp -(n - j ) t2 2n

t2 4 + 3 t4 + 2n 24 n

exp -(n - j )

1+ t2 2n

4 + 3 t4 exp 24 n 1+

4 + 3 t4 24 n
/2

exp -(n - j ) , Bk
,n,j

4 + 3 t4 1 e 24 n

.

Bk

n n-j

(k+1)/2

1+

4 + 3 e (k + 3)(k + 1) 24n
1/4

n n-j

2

,

12 Bk,n,j T = 4 +3 n-1/4 . (18) T . , (20) 6n 0 < t 4 +3 . n/h. , 46 < . , +3 h

h2 < 2 /64 + 3. [10] , h 1 + 3 , , 3 4 . 1 1, ,
2

1+

4

<

2 (4 + 3) , 6

. , (18) f t n
n

exp -

t2 4 + 3 t4 + 2 24 n

e exp -

3n 4 + 3

,

n e exp - h

3n 4 + 3

.

, (19) OL 1 n 4 + 3 1 B4 + 4!n 2 3n 4 + 3 3 3! n
2

B6 +

|3 | 1 |3 | 4 + 3 B+ B7 + 3/2 5 12n 2 3! n 4!n
1/4

n + e exp - h

1 + 1/4 n

12n 4 +3

4 + 3 12
1/4

exp - dt,

3n 4 + 3

+

|3 | 1 + 3! n

t3 e

-t2 /2

172

.. . . .-.

|f (t)|. P , 2 |f (t)| e-t /2 |t| T /h. , (13) t Tn f n n - g (t) ih e-itx dt+ Fn (x) - G(x) = 2 2 n -T n sin(th/2 n) ih + 2 2 n
T n

e-itx g (t)
n

-T

1 1 - sin(th/2 n) th/2 n e-
itx

dt+

+

1 2

|t|> n/h

g (t) dt+ it

ih + 2 2 n

T

n<|t| n/h

e

-itx

ih dt - 2 2 n sin(th/2 n) e
-itx

f

n

t n

T

n<|t| n/h

e

-itx

g (t) dt+ sin(th/2 n) dt.

+

ih 2 2 n

T

n<|t| n/h

g (t)

1 1 - sin(th/2 n) th/2 n)

1 3 1 - x 6 2 n
2

e

-x2 /2

, Th (9) /2 sin(T /2 2) , , h/ (9), , , T n. , (9) , n , n (T ) ln , 2 Th 1- e 2T 2 n
T 2 n/2

1 12

h 2n

2

e-

T 2 n/2

.

, (9) , n , . , T , 1.55, T = 0.25 T h/2 sin(T h/2) < 1.011, . . (9), n , , . , T : T n , , n . . . (1) (3) . , . ,

l =
-

Hl (x)P dx,

3 = 3 4 = 4 - 3. 173

2. f f
n

n

t n

- g (t) n 3 -

t n it n
3

3 - g (t) = 3!
6 2 Cn g n-2

it n

3

ng

n-1

t n

4 + 4! t n

it n

4

ng

n-1

t n

+

+

3 3!

2

t n

+

4 4 tµ 4!n

n-1

3 4 + 3 7 + tµ 3! n 4!n

n-2

t n

+

6

3 3! n

t9 µ

n-3

t n 6

+

6 tµ 48n2
2

n-1

t n
n-3

+

4 4 + 3 8 tµ 2 4!n 4!n t n .

n-2

t n

+

3 3! n

4 + 3 10 tµ 4!n

(21)

1, f (t) g (t) , f (t) = 1 - t2 3 4 4 + (it)3 + (it)4 + t4 , 2 3! 4! 4! g (t) = 1 - t2 t4 t6 + + , 2 8 48

- < t < , , f t n -g t n 3 = 3! it n
3

4 + 4!

it n

4

4 + 4!

t n

4

1 + 48

t n

6

.

, 4 4 , - . 1, , f
n t n

- g (t) =

3 3!

it n

3

+
4

4 4!

t n

4

n-1 j =0

f

n-j -1

t n 6

g

j

t n

+ t n

+

4 4!

t n

nµ

n-1

t n

+

1 48

t n

nµn

-1

.

3 3! +
3 3! it n 3

it n

3

+

4 4!

it n

4

ng

n-1

t n
2

+

+

4 4!

it n

4

n-2 j1 +j2 =0

f

n-j1 -j2 -2

t n

g

j1 +j

t n

f

t n

-g

t n

.

f g 1, , 3 3!
2

it n

6

n-2

f
j1 +j2 =0

n-j1 -j2 -2

t n

g

j1 +j

2

t n

+

3 4 + 3 3! 4!

t n

7 2 Cn µ n-2

t n

+

174

.. . . .-.

+

3 4 3! 4!

t n

7 2 Cn µ n-2

t n

+

4 4 + 3 4! 4!

t n

8 2 Cn µ n-2

t n

.

, 3 3! + 3 3!
2 2

it n

6 2 Cn g n-2

t n

+ t n t n t n

it n

6

n-3

f
j1 +j2 +j3 =0

n-j1 -j2 -j3 -3

t n

g

j1 +j2 +j3

f

-g

,

j1 , j2 , j3 3 , 0 j1 + j2 + j3 n - 3. Cn . 3 3!
3

t n

9

C

3 n-3 nµ

t n

+

3 3!

2

4 + 3 t 10 3 Cn µ 4! n

n-3

t n

.

, , |4 | 4 + 3. , , 2 x Dn 3 4 1 pn (x) - (x) = H3 (x)(x) + ~ H4 (x)(x) + 4!n 2 3! n OL 1 n 4 +O 8 2 n 1 n
3/2

3 3! n

2

H6 (x)(x) + OL

1 n

,

.

(22)

1 O n3/2 , , n . (21). t t t (21) g n-1 n g n-2 n g n n . , , (21)

1 2

e
-

-itx

3 3!

2

it n

6 2 Cn e -t2 /2

dt =

1 2

3 3! n

2

H6 (x)(x) -

1 2n

3 3! n

2

H6 (x)(x).

1 pn (x) - (x), O n3/2 . ~ 2 (13) ¯ : x Dn 3 4 1 Fn (x) - G(x) = - H2 (x)(x) - H3 (x)(x) - 4!n 2 3! n 1 + 6 h 2n
2

3 3! n ,

2

H5 (x)(x)+

H1 (x)(x) + O 175

I

1 n

1 1 4 +O . 3/2 n 12 n n OI 1 1 u - sin u u - (u - u3 /6 + u5 /120) u2 u4 -= = = + sin u u u sin u u sin u 6 sin u 120 sin u u u2 u2 =+ 6 sin u 6 6 1 1 - sin u u u u2 u - sin u u u2 u3 /6 u u4 =+ =+ = + . 6 6 u sin u 6 6 u sin u 6 36 sin u (23)

(23) (9). (9) , n 1/n. , /2 , 1. Fn (x) - G(x) , , (9) Fn (x) - G(x) . , p, 0.5, 4 . , (22) (23) 5 5 , 3/2 15 n 40 2 n3/2 . , pn (x) - (x). P n = 64. P64 ~ 1 4 - 10 p . {xk = -8 + k : k = 0, 1, . . . , 64}, q = 1 - p, p = pq 8 pq 2 , p64 (x) x = x27 = ~ 0.048869 ( ), 27 P64 , 8 pq C64 q 37 p27 = 0.396470. (x27 ) = 0.398466, p64 (x27 ) - (x27 ) -4 · 10-4 , ~ , , H4 H6 , -0.001606. p64 (x27 ) 0.396459, ~ 10-5 . , 4 . , , , x27 , H3 (x) . x23 = -0.964563, p64 (xk ) - ~ (xk ), p64 (x23 ) = 0.254593, (x23 ) = 0.250541, ~ 0.003427, 0.000616. p64 (x23 ) ~ -5 0.254584, 0.9 · 10 . 5.5 . x21 = -1.471280, 3 4 1 pn (x) - (x) - H3 (x)(x) - ~ H4 (x)(x) - 4!n 2 3! n 176 3 3! n
2

H6 (x)(x),

.. . . .-.

p64 (x21 ) = 0.137308, ~

(x21 ) = 0.135163,

0.001138, 0.000953. p64 (x21 ) 0.137255, 5.3 · 10-5 . ~ 5 , n = 64 15n3/2 -5 5 · 10 . , , 5 , 15n3/2 . 3 64 > 1.048, , 63 , , 2, 3 3 , , 6.4 · 10-6 .

[1] .., - / . . .: , 1999, . 364. [2] .., .., . .: , 1965. [3] ., . .: , 1947. [4] .., , . 1. .: , 1954. [5] .., .., . . . . .: , 1967. [6] .., : . .: , 2009. [7] .., // . ., 2014, . 59, 2, . 276312. [8] .., , // . . ( ). [9] .., . I, II, III. // . ., 2001, . 46, 2, . 326344, 2001, . 46, 3, . 573579, 2001, . 47, 3, . 475497. [10] .., // . . 2009, . 3, 3, . 6978. [11] .., // . , 2012, . 443, 5, . 555560. [12] Uspensky J., Introduction to mathematical probability. N.-Y.: McGraw-Hill, 1937.

177

1
..2 , ..
3

, , , . " " . , " " , , .. , .

1

() , -, , , , . : a) ; b) . . , , [2]. . ,
.. " " - - . 2 , echepurin@mail.ru, , . .. . 3 , ni.ta@bk.ru, , - . .. .
1

178

.., ..

, , , . , . , , , . , . , : , . , . 1964 , .. , , , . [8], [7] ., [4], [5] [6]. , CH- . , , . CH- . CH- . , CH- . , , , , .. , , . : . y = (y1 , . . . , yn ), yi Rk , yi (...) , i = 1, n, L(y1 ) = P{y1 < u}, u Rk . Fj (u; j ), j , j j , j , j = 1, b. A1 : 11 : L(y1 ) = F1 (u; 1 ), 21 : L(y1 ) = F1 (u; 1 ), A2 : 12 : L(y1 ) = F1 (u; 1 ), j : L(y1 ) = Fj (u; j ), 22 179 1 1 , j j , j = 2, b. 1 1 , 1 1 .

B: . , . , Gq (u; q ), q q , , . , y = (x, w), x w , x = (x1 , . . . , xn ), xi Rk , xi ... , i = 1, n, L(x1 ) = G1 (u; 1 ), u Rk , 1 1 , w = (w1 , . . . , wm ), wj Rk , wj ... , j = 1, m, L(w1 ) = G2 (u; 2 ), 2 2 , u Rk . B : 3 : G1 (u, 1 ) = G2 (u; 2 ), 4 : G1 (u, 1 ) = G2 (u; 2 ), u Rk ; q q , q = 1, 2. u Rk ; q q , q = 1, 2,

C: . y = ((x1 , w1 ), . . . , (xn , wn )), xj Rk1 , wj Rk2 , (xj , wj ) ... , j = 1, n, L((x1 , w1 )) = K(u1 , u2 ), L(x1 ) = K1 (u1 ), L(w1 ) = K2 (u2 ), u1 Rk1 , u2 Rk2 . C : 5 : K(u1 , u2 ) = K1 (u1 )K2 (u2 ), 6 : K(u1 , u2 ) = K1 (u1 )K2 (u2 ), u1 Rk1 , u2 Rk2 . u1 Rk1 , u2 Rk2 .

CH- . - .

2
2.1

CH- A
CH-

CH- . , . , y , , 1 2 . , , n, , , , . (v ), y . v , v V , V v . (v ) CH-. CH- , , , , .. , (. [13]) (v ) , (v ). 180

.., ..

(v ) . . 0 0 K 1 2 . , (v ) 1 1 (v , 0 ), 1 , .. 2 , CH- (v ) 2 (v ; 0 ). 1 (v , 0 ), 2 (v ; 0 ) 0 0 . , y n (v ). , rn (v ), , 1 . , ^ , , rn (v ) , ^ y 1 . n , 1 . , tn (y ) 0 1 , , , 1 (v ; tn (y )) = 1 (v , 0 ) + 1n (1) j n , ..
n

lim P{||j n || > } = 0,

j = 1, 2, . . . ,

> 0.

, 1 , 2 , , > 0
n

lim P{||tn (y ) - m(0 )|| > ; 0 } = 0,

> 0,

(2)

.. tn (y ) 2 m( ) = 0 , 1 (v ; tn (y )) = 1 (v ; m(0 )) + 2n . (3) , rn (v ) = ^ 1 (v ) + 1n , 1 , 1 (v ; m(0 )) + 2n , 2 . (4)

(v ) 1 2 , , ^ "" 3 (1 2 ). Rn (v ). n ^ Rn (v ) = 1 (v ; 0 ) + 3n , 1 , 2 (v ; 0 ) + 4n , 2 , (5)

^ .. Rn (v ) (v ). , Ror , : ^ R = Rn (v ), v V , r = rn (v ), v V . ^ (6)

n (6) R = r, 1 R = 2 (v ; 0 ), v V , (7) r = 1 (v ; m(0 )), v V , 181

2 . , (7) v r. (7) R = 2 (-1 (r; m(0 )); 0 ), 1 (8)

-1 1 . 1 , , (8) (7). 2 (v ) m(v ) . , (4) (5) , 1 (6) R = r. 1 . 2 (6) (8). 2 . (8) R = r, 1 2 . , : . (6) R = r, 1 . (6) R = r, 1 2 . , 2 , , (8). (8) , 2 . (6), , , , , y . , , . , . . (6) , . , (6) ^ ~ ^ R = n Rn (v ) - rn (v ) , r = rn (v ), v V , ~^ v V,

~~ (R, r). (6), (6) R = r. , (6) R = r. . , , . CH- . , . 182

.., ..

2.2
2.2.1

"PP"
A. . } , v R1 . : P{y1 < v } = F1 (v ; 0 ), 0 1 , : P{y1 < v } = F1 (v ; 0 ), 0 1 , : P{y1 < v } = F2 (v ; 0 ), 0 2 ,

k = 1, (v ) = P{y1 < v , 1 2 2

(9)

Fi , i , 0 0 , ^ y , , 1 2 . n 0 1 , (6) 1 2 , 1 2 ^ R = Fn (v ), ^ r = F1 (v , n ), ^ Fn (v ) =
1 n n

v R1 , v R1 ,

(10)

1 yi < v ) . I(
i=1

1 A) = I(

1, A , 0, A ,

A. (10) - . "PP" , "PP" . , 1 . , 1 . [4], , "PP" . 1914 . 1 2 . (10) v y(1) v y(n) . ^ Fn (yi ) : i , n i , n+1 i + 0.5 , n i - 0.5 , n i - 0.325 . n + 0.25

(10) . 1 2 . 1. . 1 : F1 (v ; 0 ) = ln v - µ0 2 , 0 = (µ0 , 0 )T , 0 2 - < µ0 < , 0 > 0, v > 0. 183

u

(u) =
-

1 2

exp{-t2 /2}dt -

. , v0 2 : F2 (v ; 0 ) = 1 - exp - , 0 0 = (0 , 0 )T , 0 > 0, 0 > 0, v > 0. "P-P" ^ R = Fn (v ), ^ r = ln v^-µn , n n 1 µn = 1 ln yi , n = n (ln yi - µn )2 , ^2 ^ ^ n
i=1

(11)

^ n = (µn , n ) 0 1 . 1 ^ ^2 , (11) n R = r. 2 , µ ^
n

- E{ln y1 ; 2 } = µ, ~
p

p

n - D{ln y1 ; 2 } = 2 . ^2 ~ , µ = ln 0 - ~ ~
2

, 0

= - (1) , = 0.57721 . . . ,

=

2 . 2 60

, 2 (10) n R = 1 - exp - r=
ln v -µ ~ ~ v 0 0

, v > 0, v > 0.

(12)

,

(12) ln v , (12) 1 ln ln 1-R - , 0 < r < 1. r =
2 6

(13)

R , r , (13) , (. . 1) R = 1 - exp - exp + ur 2 6 , (14)

ur = -1 (r) r . , 184

.., ..

. 1: "P-P" . Dmax = 0.234621 r = 0.485, R = 0.816. 2 (12) (13). 1 2 , 2.1. , A2 "P-P" (7), j , j = 2, b, 22 . A1 . "PP" . , , , , , "P-P" . (. 2.3). , ^ "PP" , F1 (v , n ) v R1 . , , (. [3], [1], [10], [9]). , CH-, . . 185

2.2.2

. .

. , , "P-P" 1 2 2 k > 1. , ^ , F1 (v , n ) k v V R . , CH-, . , , , . , yi = (yi1 , yi2 , . . . , yik )T ... k - , i = 1, n, v = (v1 , . . . , vk )T Rk , k > 1, () F1 (v, 0 ) F2 (v, 0 ) v 0 , 0 . ^ , (9) F1 (v, n ) k 1 F1 (v, 0 ), v R . ^ ^p n 0 1 n (0 ) 2 , (0 ) , F1 (v, (0 )) . (10) ^ R = Fn (v), (15) ^ r = F1 (v, n ), v Rk , ^ Fn (v), , , 1 ^ Fn (v) = n
n

1 Wi (v)), I(
i=1

Wi (v) = {yi < v} = {k=1 {yij < vj }}. j (15) , (10), . , (15) r . , r = 0 r = 1. , ^ Vc F1 (v, n ), .. ^ Vc = {v : F1 (v, n ) = c, v1 , v2 Vc , v1 = v2 , P{W1 (vi )/(W1 (v1 ) W1 (v2 )); 0 } > 0 ^ ^ P{Fn (v1 ) = Fn (v2 ); 0 } > 0. , ^ ^ P{min Fn (v) < max Fn (v); 0 } > 0.
v V
c

0 < c < 1}.

i = 1, 2,

v V

(16)

c

(16) , r = c R (15) . , "P-P" (15). n (15) R = r 1 R = F2 (v , 0 ), v Rk , r = F1 (v , (0 )), v Rk , 186 (17)

.., ..

. 2: (18) . 2 . , (15). v = yi , i = 1, n. "P-P" (ri , Ri ), ^ Ri = Fn (yi ), (18) ^ ri = F1 (yi , n ), i = 1, n, (15), . v = yi , F1 (v , 0 ). 2.2.1 ^ Fn (yi ) . "P-P" . 2. . ... n = 200 µ = (1, 5)T = 43 38 .

, (-) 500 (. 3). (. 187

[11]), . . 2 (18), . . , , -, , . , .. P{y1 > v; 0 }, ¯ ¯ . , " P - P" , n R= 1 1 yi > v ), v Rk , I( + n i=1 ^ r = P{y1 > v ; n }, v Rk . + - , , [12].

2.2.3

B. .

"P-P" ^ R = G2 (u), u U ^ r = G1 (u), u U ,

(19)

n+

^ Gi Gi , i = 1, 2. k = 1 U R1 , k > 1 U = {uq , q = 1, n + m : ui = yi , i = 1, n, ui = wj i = 1, n + m, j = 1, m}.

" - " .

" -" n = 200, µ1 = µ2 = (1, 5)T , 1 = 2 = 43 . 38

188

.., ..

" -" n = 200, µ1 = (1, 5)T , µ2 = (6, 9)T , 43 1 = 2 = . 38

" -" n = 200, µ1 = (1, 5)T , µ2 = (-2, -4)T , 43 1 = 2 = . 38

" -" n = 200, µ1 = (1, 5)T , µ2 = (3, 7)T , 72 43 1 = , 2 = . 38 29

2.3

" - ".

y = (y1 , . . . , yn ), yi Nk , .. yi = (yi1 , . . . , yik ), yij 0, yij , + j = 1, k , yi ... , i = 1, n, z = (z1 , . . . , zk ), zj , |zj | 1, j = 1, k , L(y1 ) = F0 (u; 0 ), u Nk , +

L (y1 ) = f0 (u; 0 ) u Nk . + y1
k

(z ; 0 ) = E
j =1

zj 1j ;

y

0

(20)

189

, ,
k

(z ; 0 ) =
(m1 ,...,mk )N
k +

mj zj j =1

f0 ((m1 , . . . , mk ); 0 ) .

(21)

. [13], " ". (z ; 0 ) , z = (z1 , . . . , zk ) , zj 0 zj 1 j = 1, k . 1 ^ (z ) = n
n k

zi
i=1 j =1

y

ij

(22)

^ (z ; 0 ) n . , n ^ n 0 , (z ; n ) (z , 0 ). 2.3.1 A.

y1 Nk . A1 A " - " + ^ R = (z ), z = (z1 , . . . , zk ), 0 zj 1, j = 1, k , ^(1) r = (z ; n ), z = (z1 , . . . , zk ), 0 zj 1, j = 1, k ,
(1)

(23)

^ n 0 1 . " " 2.1. . " " "-" ,
(1) ~^ R = n{(z ) - (z ; n )}, (1) r = (z ; n ), 0 z 1. ~

0 z 1,

(24)

(23), 1 . , "-" . ^(q) A2 A n (q ) 2 , , (23), R = (z ), (25) ^(q) r = (z ; n ) q = 2, b. , " - " . , A1 . 3. . y = (y1 , . . . , yn )T , yi ... , i = 1, n. 190

.., ..

1. , 1 : y1 = P OI S (0 ),
d

0 > 0,

L (yi ) =

k 0 e-0 , k!

k = 0, 1, 2, . . .

(26)

(z ; P OI S (0 )) = exp(0 (z - 1)) a1 (P OI S ()) = yn = 0 + op (1) ¯ n R = (z ) = 1 ^n z yi = E {z yi ; P OI S (0 )} + op (1) = exp(0 (z - 1)) + op (1) n i=1 r = (z ; P OI S ()) = exp{yn (z - 1)} = exp(0 (z - 1)) + op (1) ¯ R = r + op (1) n .

(27)

(28)

(29) (30)

, 1 (r, R) . 2. , (r, R), 2 : y1 = N B I N (0 , 0 );
d

0 > 0,

0 < 0 < 1,

(31)

L (y1 ) =

(0 + k ) 0 0 (1 - 0 )k , (0 )k !

k = 0, 1, 2, . . .

(32)

(z ; N B I N (0 , 0 )) = =

0 0 = 1 - (1 - 0 )z 0 0 - (1 - 0 )(z - 1)

(33)
0

,

|z | <

1 . 1 - 0

" - " R 2 1 ^ R = n (z ) = n =
n

z yi = E {z yi ; N B I N (0 , 0 )} + op (1) =
i=1

0 0 - (1 - 0 )(z - 1)

0

+ op (1),

(34)

a1 (N B I N (0 , 0 )) =

0

1 - 0 , 0

(35)

a1 (N B I N (0 , 0 )) = yn = ^ ¯ 191

0

1 - 0 + op (1). 0

(36)

" - " ^ r =(z ; P OI S (a1 (N B I N (0 , 0 )))) = exp ^ exp - (37) (1 - 0 )(z - 1) = 0 ln r + op (1) 0 (39)
0 0

1 - 0 (z - 1) + op (1), 0

(37)

1- 0

0

r 1.

(38)

(34) R= 1 1-
0 ln r 0

+ op (1), exp -

0

1 - 0 0

r 1.

(40)

, 2 (r, R) , (40), . 3.

. 3: " - " . , " - " y (40). 192

.., ..

" " (. 2.2) (40) (41) ~ R = exp{0 (1 - R r=r ~
-
1 0

)}

(41)

, . n (41) 0 0 n n ^ ^ " - " y , 2 .

[1] Fama E.F. , The Bevavior of Stock-Market Prices // Journal of Business, XXXVIII, (January, 1965), p. 34105. [2] Anscombe F.J. , Graphs in statistical analysis // American Statistician, 1973, v. 27, p. 1721. [3] Mandelbrot B., The Variation of Certain Speculative Prices // Journal of of Business, XXXVI, 1963, p. 394419. [4] Fisher N.I., Graphical methods in nonparametric statistics: A review and annotated bibliography // Int. Statist. R., 1983, v. 51, p. 2558. [5] Cleveland W.S., Research in Statistical Graphics // J.A.S.A, 1987, v. 82, 398, p. 419 423. [6] Huber P.J., Experiences with Three-Dimensional Scatterplots // J.A.S.A, 1987, v. 82, 398, p. 448453. [7] du Toit S.H.C., Steyn A.G.W., Stumpt R.H., Graphical Exploratory Data Analysis. Springer-Verlag, 1987. [8] .., STATISTICA. . " ". .-, . . ., 2001, . 220. [9] .., . ., , 1983. [10] .., . . ., , 1998. [11] .., // , . . ., 1995, . 1, . 112125. [12] .., // . . " ", 1994, . 1, 2, 279330. [13] ., . ., , 1984, . 1.

193

..
1

[1] . () . (). N. , . , . .

. . i- Ai Ai , Ai Ai , Ai , i = 1, 2, . . . P(A1 ) = P(A2 ) = . . . () N . Ai , [0, t0 ] i- , i = 1, 2, . . . , N . , Ai . R = P(A1 ) = . . . = P(Ai ) = . . . () [0, t0 ]. t0 . , , , , , (. [2, . 21.2, 21.5], ). . T t0 . N (). T = t0 N . [1] .. , " N ". (.. ) .. Ai , i = 1, 2, . . ., . [1] (. 1.2.2) (. 2.3)
, nchist60@yandex.ru, , , - . .. .
1

194

..

(). , , , . , [1] . 1. .. [1, . 1.22, . 40], . R = P(Ai ), Ai i- , i = 1, 2, . . . , N , .
k

P(N = j ) = Rj (1 - R),

P(N k ) =
j =0

P(N = j ) = 1 - R

k+1

,

(1)

, N . [1] N , ( ) " : 1) N N , N , 2) N = N , N l(N ) = 0, 3) N = N , l(N ) = 0." , N = min(N , N ), (2)

N , N . N . G(k ) = P(N k ) = P(min(N , N ) k ) = 1-R 1
k+1

k < N k N

.

(3)

- .. N . , F (k ) = P(N k |N N ) = P(N k ) P(N N ) = 1 - Rk+1 . 1 - RN +1 (4)

, k N , k > N 1. C , F (k ) N . N , N [1] , . ( 1.9, [1, c. 41]): " , ( ), : . . i- : ( Ai ) ( Ai ). 195

1,05

1

0,95

0,9

0,85 G (X) 0,8 F(X) 0,75

0,7

0,65

0,6 20 22 24 26 28 30 32

. 1: F (k ) G(k ). . : 1) , l(N ) = 0; 2) N , l(N ) = 0. c N , , l(N ) = 0 N = N l(N ) = 0. . N N (R), .. Ai i = 1, 2, . . . R = P(Ai ), , . N F (1.81). . . . " "" . Ai , i = 1, 2, . . ., , . , , , [1] . . (1.81) [1] c (4). . (1.81) : P(N k ) = 1 - : 1 - Rk+1 = F (k ), 1 - RN +1 , , N , (3). F (k ) G(k ) . . 1, F (k ) G(k ) R = 0, 95 P(N k ) = 196 1 - Rk+1 , 1 - RN +1

..

N = 30. R N . N , (4). , , N N , .. (N + 1) - . 1. N N . N [1] . [1] N N . N [1] . 2. , .. , (.. N ), N (4). [1, . 2.3, . 98] . (1.81) [1] 2.17 2.18. . , , [1] , N N , .. , (4), (1.81). (1.81) [1] . 2.17. . R = 1 - xN 1 - xN

def

+1

= 0 (x),

0 = 1 -

1 , N +1

(5)

= 0 , N , N x [0, 1], c R > 0 . N = 0 R = 0. . . . [1, . 7, . 50, . 1.3. ]. F (x) = P(L x) L. [0, 1], L , u = F (L) + (1 - )F (L - 1) [0, 1]. L = N . , u = u(N

, ) =

1 - RN 1-R

+1

N +1

+ (1 - )

1 - RN 1 - RN

+1

[0, 1]. P(u < ) = P(u ) = . , u 1 - RN 1 - RN

+1

;

1 - RN 1-R

+1

N +1
+1

. < .

, , u < , , = P(u < ) P( >

1-RN 1-RN

1 - RN 1 - RN 197

+1

) = P(0 (R) < ).

0 (x), (5), x [0, 1], x = 1 , 0 (x) = lim 1 - xN 1 - xN

x 1

+1

=

N . N +1

N N , 0 (1) NN , NN 0 (R) = +1 +1 , R . R R NN . +1 , P(0 (R) < ) = P(R > -1 ( )) = P(R > R ). 0

2.17 , N , N . [1] . 2.18. 2.17 , [0, 1] R R R = (1 - )(1/N ) , N = 0 R = 0. R R . . N R = 0 (R ) R , 1 - (R )
N

= 0 R = 0 . N = 1 - (R )
N

> 0. (6)

0 (R ) 1 - (R )

N

,

R R . 2.17 : P(R > R ) P(R > R ). T 2.18 . [1] R ( 2.17) R ( 2.18), R R . . N = 9 = 0 = 1 - 1/(N + 1) = 0, 9. 1 2 N R R 0 0 0 3 0,46 0,48 5 0,63 0,65 7 0,72 0,78 8 0,75 0,86 9 0,77 1

, 2 , N = 9 N = 9 (.. ) = 0, 9. , . 198

..

c 2 . : 2.17 , .. N ? : 2.17 N , N , , 2 ? L = N , .. , (3). , u = u(N , ) = (1 - R
N +1

) + (1 - )(1 - RN )

. = 0 (R) 0 (R) = 1 - RN . R (N ) R (N ) = (1 - )(1/N ) R (N ) . 2. , 2.17, N , , (3), , . 2.18 R = (1 - )(1/N ) , 1 . 2 . , 2.17 , , , (4), N . , N (5). , N , N = N , N . , N = N , N + 1 , N (N + 1)- . 3. N = N , , 2 , . . , [1] , (). , , [1]. , [1].

[1] .., . . .: , 1988. [2] . . .: , 1985.

199

, . , . "..." " ". 1. , , , ", " ( ). ? 2. A B n, trA > 0 trB < 0. , x Rn , (Ax, x) > 0 (B x, x) < 0.
n n 3. (X1 , . . . , Xn ) n {x Rn : xi 0, i = 1, . . . , n, i=1 xi = 1}, n N. , n {nX1 }nN n , .

4. S0 = 0, Sn = X1 + . . . + Xn , n N, {Xi }iN ... , P(Xi = 1) = 1 - P(Xi = -1) = p > 1/2. {Sn , n Z+ }. 5. ( ). n > 3 ( , , ; ). , . 6. (.. ). , S . 7. Rn , . ) E 2 ; ) E 4 . 8 (3-5). X , p , .. C > 0 , p C (: X N (0, 1)). . 9. ... X Y , t > 0 P(|X + Y |/ 2 t) P(|X | t). , ) ; ) (3-5) . 200

Abstracts

Abstracts
Location of p ositive eigenvalues in the spectrum of the evolutionary operator for a branching random walk
Antonenko E.A., Ph.D. student, Faculty of Mechanics and Mathematics, MSU, e-mail: eka.antonenko@gmail.com Yarovaya E.B., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: yarovaya@mech.math.msu.su
We consider a continuous-time branching random walk with finite variance of jumps on a multidimensional lattice with a finite set of the particle generation sources. The structure of the positive discrete spectrum of the evolutionary operator for a branching random walk with finite variance of jumps is investigated.

Queueing systems with regenerative input flow
Afanasyeva L.G., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: afanas@mech.math.msu.su
The aim of this article is to demonstrate possibilities of the renewal theory techniques for asymptotic analysis of queueing models. The article is of the survey character. We begin by definition of regenerative flow and discussion of its properties. Then equivalence of the stability and stochastic boundedness for the broad class of processes describing queueing systems with regenerative flows is considered. Basing on the results obtained the ergodicity conditions for multichannel queueing system with heterogeneous servers are given. The next two problems under consideration are convergence rate in total variation to limit distribution and estimates for large deviations probabilities. Some examples are also given.

Asymptotic b ehaviour of some sto chastic storage systems
Bulinskaya E.V., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: ebulinsk@yandex.ru Sokolova A.I., student, Faculty of Mechanics and Mathematics, MSU, e-mail: alun@mail.ru
The aim of the paper is investigation of a system with seasonal demand and/or replenishment of the stored product. It is supposed that the system is described by a generalized renewal process. We study several types of generalization of the simple renewal process widely used in various applications of probability theory. The strong law of large numbers and central limit theorem are proved for the processes under consideration as well as the analog of the elementary renewal theorem with estimation of convergence rate. The results are used for optimization of storage system performance.

201

Abstracts

Conditional b ounds of risk measures in financial mathematics
Grigorieva M.A., student, Faculty of Mechanics and Mathematics, MSU, e-mail: 011235813@inbox.ru
We found the bounds of risk measures VaR, CVaR and MINVaR for standardized random losses under some additional conditions.

Exp onential utility maximization in a L´ evy model
Ivanov M.Yu., researcher, Faculty of Mechanics and Mathematics, MSU, e-mail: m_y_ivanov@mail.ru
We consider the exponential utility maximization problem in an exponential L´ model. evy Under certain restrictions on the set of wealth processes it is possible to apply to the main problem the same methods which were applied in logarithmic and power cases, and also avoid additional restrictions on the asset price process. We find the solution which belongs to the class of admissible strategies, and is not only approximated by a sequence of wealth processes like in many well-known papers. At the end of our work the dual problem is considered.

Intermittency and product of random matrices
Il larionov E.A., PhD student, Faculty of Mechanics and Mathematics, MSU, e-mail: illarionov.ea@gmail.com Sokoloff D.D., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: sokoloff.dd@gmail.com Tutubalin V.N., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: vntutubalin@yandex.rum
Systematic elaboration of a phenomenon of intermittency, i.e. a progressive growth of higher statistical moments of a vector field in a random medium, has been initiated by Ya.B. Zeldovich in the context of astrophysical and magnetohydrodynamic problems. At that time the mathematical aspects underlying the physical description were still under development and connection between different problems was rather unclear. Contemporary results from the theory of the product of independent random matrices (the Furstenberg theory) give a new way of tackling of the phenomenon of intermittency. We consider several examples and discuss the approach to investigation of multiplicative random variables.

Optimization and limit distribution in a discrete insurance model
Karapetyan N.V., e-mail: karanar@mail.ru
Discrete time model of insurance company is considered. It is supposed that the company applies a dividend barrier strategy. The limit distribution for the time of ruin normalized by its expected value is found. We assume that shareholders cover the deficit at the time of ruin and investigate barrier strategies maximizing shareholders' dividends and profit accumulated until ruin. In case the additional capital is injected right after the ruin to enable infinite performance of the company, existence of optimal strategies is proved both for expected discounted dividends and net profit.

202

Abstracts

Extremes of dep endent scores of particles in branching processes
Lebedev A.V., associate professor, Faculty of Mechanics and Mathematics, MSU, e-mail: avlebed@yandex.ru
We consider branching processes in which each particle has some random score with a given distribution. It is supposed that scores of particles are dependent and this dependence is associated with their kinship distance. We are interested in the asymptotic behavior of scores extremes in generations. The nondegenerate limit laws for maxima are established under linear normalizations.

Archimedian operations and copulas with applications to branching pro cesses with interaction of particles
Lebedev A.V., associate professor, Faculty of Mechanics and Mathematics, MSU, e-mail: avlebed@yandex.ru
Generalized summation operations (commutative and associative binary operations) which are isomorphic to the addition are considered in this paper. We establish the correspondence between operations and strictly Archimedean copulas and their properties. The possibility of application to branching processes with competition-like interaction of particles is also investigated. We discuss the "Archimedity" concept applied to operations and copulas, as well as its possible extension to stochastic operations.

New models of transp ort flows dynamics
Lykov A.A., Ph.D., Faculty of Mechanics and Mathematics, MSU, e-mail: alekslyk@yandex.ru Malyshev V.A., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: 2malyshev@mail.ru Melikian M.V., student, Faculty of Mechanics and Mathematics, MSU, e-mail: magaarm@list.ru
We find stability conditions of the one-dimensional car flow where motion of each car is determined only by the motion of the previous one.

On the two results of J. Usp ensky and one result of C.-G. Esseen
Senatov V.V., professor, Faculty of Mechanics and Mathematics, MSU, e-mail: v.senatov@yandex.ru
We discuss the J. Uspensky's results obtained in 1937 that give asymptotic expansions with explicit estimates of residuals for binomial distributions. A simple consequence of one of them is the lower bound, obtained by Esseen in 1956, for the constant in the Berry Esseen theorem. The Uspensky's results are generalized to arbitrary lattice distribution and improved at the same time. It is shown how to change the statement of the problem of approximation accuracy in the CLT for lattice distributions to obtain estimates that are many times more accurate than that given by the BerryEsseen theorem.

203

Abstracts

Graphical methods of exploratory data analysis
Chepurin E.V., associate professor, Faculty of Mechanics and Mathematics, MSU, e-mail: echepurin@mail.ru Nifontova T.A., engineer, Faculty of Mechanics and Mathematics, MSU, e-mail: ni.ta@bk.ru
This report is devoted to visual analysis methods of distributional type for one-dimensional and multivariate discrete. The testing is based on a CH-probability plots. estimated plot from the standard plot defined by a null hyp statistic. hypotheses testing about a data, both continuous and A departure quantity of an othesis is considered as a test

On planning the scop e of the control tests based on stopp ed binomial failure test scheme
Chistyakova N.V., senior scientific researcher, The Laboratory of Probability Theory, Faculty of Mechanics and Mathematics, MSU; e-mail: nchist60@yandex.ru
In monograph "Technical Systems Testing. Evaluation Values and Durations" by R.S. Sudakov the Stopped Binomial Scheme (SBS) of failure test is determined. According to SBS no more than N identical prototypes are tested, but the trials are stopped after the first failure. The advisability (expediency) of such a scheme is substantiated by the possibility of sizeable reduction of the number of trials required for adjusting of prescribed level of reliability in comparison with the number of such trials required by BS. We investigate the ossibility of using SBS to confirm the requirements on reliability of technical systems in case of failure-free tests.

204

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 .., .., . . . . . . . . . . . . . . . 9 .., . . . . 23 .., .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 .., . . . . . . . . . . . 63 .., . . . . . . . . . . 83 .., .., .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 .., . . . 121 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 .., .., .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 .., . . .-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 .., .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 . . 200

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 205

1

I. 1. .. , .. . 2. .. , .. . I I. 1. .. , .. . 2. .. , .. . I I I. 1. .. , .. . 2. . .. . 3. . .. -. IV. 1. . .. . 2. . .. , .. . 3. . .. , .. , .. . V. 1. . .. , .. . 2. . .. , .. . 3. . .. . VI. , .. . .
1

1. 105- .. . .. , .. . 2. 100- .. . .. .. 3. 100- .. . .. .. -

: http://www.math.msu.su/sb80

VI I. . 1. 190- .. . .. , .. , .. , .. . 2. 80- - .. . .. , .. . VI I I. 1. 80- .. . - . .. , .. , .. . 2. 80- - .. , 130- .. , 85- .. . .. . 3. 80- - .. 110- ... . .. , .. IX. 1. 80- - .. . . .. , .. , .. 2. 80- - .. . . .. , .. 3. 80- - .. . . .. .. X. 1. 60- . .. 2. 100- . ..

207

X. 3 80- - .. .. , ..

-: ..

15.05.2015 60 90/16. . . 13. . -34. . 117312, , - , 11, . 11. 100 .