Документ взят из кэша поисковой машины. Адрес оригинального документа : http://chaos.phys.msu.ru/loskutov/PDF/HOMODYN.pdf Дата изменения: Mon Jan 31 16:04:24 2011 Дата индексирования: Mon Oct 1 19:49:45 2012 Кодировка:

. . . .., ... , , 2007, .307­315.

1



.., ..
...,

, .. . , . , . .

517.9

2


1



. ґ , , , . , . (.. ), . , , , . , , ( ) . , (, ) {ym }. y = {ym } ^
m=0

-

, [1­3]: ) {T } M = {x}, x0 , (T m (x0 )) = y
m

m = 0, 1, 2, . . . ; )

dist(T t x, T t x ) const · et dist(x, x ), .. {T } , . , y, ^ , , , . , , . , , ( y ), ^ . ( ), . , , . (., , [4­6] ) , 3


. , . dn p =F dtn , t dn-1 p dn-2 p , , . . . , p, a dtn-1 dtn-2 ai , (1)

i

F

. -

(1). (., , [7­9] ) , F . , (1). , . : . , . . ; "" . .

2



(1) n . n- . , . (1) , .. . , (.. ) n . , , . -

4


. , ( , . ) . .

2.1



, , . 1) . 2) : · . . · , . . 3) . , n- , . , . 4) · , .. , ; , ; · , (. 3)); . , , ґ . , .

5


2.2



, : P (t) = p(t) + (t) , dn p(t) + dtn P (t) , p(t) t, am
n-1 m=0

dm p(t) am =0, dtm

(2)

t, (t)

-



.

. (2) :
n

p(t) =
m=1

zm ew

m

t

.

, zj wj .

2.3



(t) , . :
T n 2

P (ts ) -
s=1 m=1

zm e

wm t

s

- min .

(3)

(3) , zj wj , , . ,
, .. Im wj = 0, k = j , zj = zk , wj = wk .





. -

(3) . , . (3) zj , wj , . , w1 w2 , z1 z2 . . (3)
T s=1 n
s

2

P (ts ) - 2R1 e

1 t

cos(u1 ts + 1 ) -
m=3

zm e

wm t

s

- min .

(4)

6


z1 = R1 exp(i1 ), z2 = R1 exp(-i1 ), w1 = 1 + iu1 , w2 = 1 - iu1 . (4) R1 , 1 , 1 , u1 , z3 , . . . , zn , w3 , . . . , wn . k - (k [n/2]+1, [·] ) k -1 , w1 w2 , . . . , w z2 , . . . , z
2k-3 2k-3

w

2k-2

.

k - 1 z1 z2k-2 . .
P (ts ) - 2
s=1 l=1 +1

(3) :
T k -1

Rl e

l ts

n

2

cos(ul ts + l ) -
m=2k-1

zm ew

m ts

- min .

(5)

zl = Rl exp(il ), zl 1 , . . . ,
k-1

= Rl exp(-il ), wl = l + iul , w

l+1

= l - iul . k -1

(5) R1 , . . . , R , 1 , . . . ,
k -1

,

, u1 , . . . , uk-1 , z2k-1 , . . . , zn , w2k-1 , . . . , wn . n n , + 1 + 1 (3). 2 2 , zj wj .

3



(1) . , . , . , , . . , , , . . , . , 7


(. [10]). , , , , . , , [10]. . , , , . F (. (1)) . , (1) [10]. , , , .


[1] F.Takens. Detecting strange attractors in turbulence. Springer, Berlin, 1980, p.336-382. [2] F.Takens. Distinguishing deterministic and random systems. mics and Turbulence. Ed. G.I.Barenblatt, G.Iooss, D.D.Joseph. 1983, p.314-333. [3] .., ... . . . . . ..-, ., , 1989, .238-262. John Wiley and Sons, New York, ... In: Nonlinear DynaNew York, Pitman, In: Lect. Notes in Math., v.898.

[4] E.Peters. Chaos and Order in the Capital Market. 1991. [5] E.Peters. A chaotic attractor for the S&P500. [6] E Peters. Financial Market Analysis.

Financial Anal. J., 1991, No2.

John Wiley and Sons, New York, 1994.

[7] .., ... . , . . " . ." , 1996, .165-190. 8 .,


[8] M. Casdagli. Nonlinear prediction of chaotic time series. p.335-356.

Physica D, 1989, v.35, No3,

[9] M. Casdagli, S.Eubank, J.D.Farmer, J.Gibson. State space reconstruction in the presence of noise. Physica D, 1991, v.51, No1-3, p.52-98.

[10] .., ... . III. ARCH­ . . , 2004, .11, .3, .468­486. . -

9


Homodynamicity of financial time series.
A.A.Bredikhin, A.Loskutov and A.B.Sedykh
Physics Faculty, Moscow State University

Abstract An approach to the mathematical description of financial time series which possess the property of homodynamicity, is proposed. In general, the modeling of financial series requires a large enough data, that is realized only in highly rare cases. To resolve this problem, a general type of the behavior of tra jectories near equilibrium points is considered. On the basis of such an approach a mathematical model determined up to the finite number of parameters is constructed. The efficient technique of the parameter estimation is presented.

10