Frakcionális Brown mozgás által. Sztochasztikus differenciálegyenletek megoldásának közelíté. Soós Anna
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1 Frakcionális Brown mozgás által vezérelt sztochasztikus differenciálegyenletek megoldásának közelítése Babes-Bolyai Tudományegyetem
2 fbm közelítése
3 ( ) B(t) fbm, H > 1. 2 t 0 dx(t) = F(X(t), t)dt + G(X(t), t)db(t), (1) X(t 0 ) = X 0, ahol X 0 véletlen vektor R n -ben, F és G 1-valószínűséggel: (C1) F C(R n [0, T ], R n ), G C 1 (R n [0, T ], R n ); G(, t) G(, t) (C2) F(, t),, helyi Lipschitz tulajdonságú x i t t [0, T ], i {1,..., n}.
4 ( ) a B = B(t), H (0, 1) fbm közelítése: t [0,1] az (1)-t közelítése, N N dx N (t) = F(X N (t), t)dt + G(X N (t), t)db N (t), X N (t 0 ) = X 0, (2) (2)-nek helyi megoldása, mely valószínűségben (1) megoldásához tart azon az intervallumon, ahol a megoldás létezik.
5 H (0, 1) Hurst indexű frakcionális ( ) Brown mozgás (fbm) Gauss folyamat B = B(t), t 0 ( ) E B(s)B(t) = 1 ) (t 2H + s 2H s t 2H. 2 H = 1 : standard Bm. 2
6 fbm tulajdonságai [0, T ]-n Hölder folytonos γ (0, H) praméterrel lim n 0 B(t) B(s) P ω Ω : sup δ 0<t s<h(ω) t s γ = 1, t,s [0,T ] ahol h pozitív vaószónűségi változó és δ > 0. négyzetes varációja [a, b] [0, T ]-n n i=1 ( B(t n i ) B(t n i 1)) 2 = if H < 1 2, b a if H = 1 2, 0 if H > 1 2, ahol n = (a = t n 0 < < t n n = b) az [a, b] felosztása
7 Wavelet közelítés Y. Meyer, F. Sellan, M. S. Taqqu A. Ayache and M. S. Taqqu B(t) = i= k= 2 j 2 (Ψ(2 j t k) Ψ( k))ε j,k, Ψ C 1 (R) az anya függvény: c > 0 ú.h. Ψ(t) ε j,k N (0, 1) vv. c (2 + t ) 2, Ψ (t) c (2 + t ) 3
8 Tétel B N (t) = ( P I N,j = N 2 j=0 k I N,j lim sup N t [0,1] { k Z : k j 2 (Ψ(2 j t k) Ψ( k))ε j,k } 2 N+4 (N j + 1) 2 ) B N (t) B(t) = 0 = 1
9 H = 0.9 Figure: B N közelítése fbm-nek
10 fbm közelítése trigonometriai sorral Bessel függvény J ν, ν 1, 2,... a {z C : arg z < π}: J ν (z) = k=0 ( 1) k ( z ) ν+2k. Γ(k + 1)Γ(ν + k + 1) 2 ν > 1-re, J ν vaós gyökei J H : x 1 < x 2 <... J 1 H : y 1 < y 2 <...
11 Figure: Bessel függvény: J H (with ), J 1 H (with - ), H = 0.65
12 (X n ) n N és (Y n ) n N Gauss v.v. E(X n ) = E(Y n ) = 0 ahol VarX n = 2c 2 H x 2H n J 2 1 H (x n), VarY n = ch 2 = sin(πh) Γ(1 + 2H). π 2c 2 H y 2H n J 2 H (y n),
13 K. Dzhaparidze és H. van Zanten B(t) = n=1 sin(x n t) x n X n + n=1 1 cos(y n t) y n Y n, t [0, 1] -abszolút konvergens sor t [0, 1]-ben -fbm
14 H = 0.65 Figure: B N közeítése fbm-nek
15 B N (t) = N n=1 sin(x n t) x n X n + N n=1 1 cos(y n t) y n Y n, t [0, 1], (3) Tétel ( P lim sup N t [0,1] ) B N (t) B(t) = 0 = 1
16 Frakcionális integrál és derivált S. G. Samko, A. A. Kilbas and O. I. Marichev: f L 1 (a, b), α > 0 -re α-rendű bal oldali frakcionális Rieman-Liouville integrálja f -nek (a, b)-n I α a+f (x) = 1 Γ(α) x a (x y) α 1 f (y)dy a.e. x α-rendű jobb oldali frakcionális Rieman-Liouville integrálja f -nek I α b f (x) = ( 1)α Γ(α) b x (y x) α 1 f (y)dy a.e. x
17 α-rendű bal oldali Riemann Liouville deriváltja f -nek α Da+f α 1 (x)= f (x) x Γ(1 α) (x a) + α f (x) f (y) dy I α (x y) α+1 (a,b) (x) let p > 1, 0 < α < 1 f I α a+ (L p(a, b)): f = I α a+φ ahol Φ L p (a, b) D α a+f = Φ a
18 jobb oldali Riemann Liouville deriváltja g-nek Db g(x)= α ( 1)α g(x) b Γ(1 α) (b x) + α g(x) g(y) α (y x) g I α b (L p(a, b)): x α+1 dy g = I α b Φ with Φ L p (a, b) D α b g = Φ I (a,b) (x)
19 M. Zähle b f (x)dg(x) frakcionális integrálja f -nek g-szerint a Ha f a+ Ia+(L α p (a, b)), g b I 1 α b (L q(a, b)), p q b a b f (x)dg(x) = ( 1) α a Da+f α a+ (x)d 1 α b g b (x)dx +f (a+)(g(b ) g(a+)) f a+ (x) = f (x) f (a+), g b (x) = g(x) g(b )
20 Sztochasztikus integrál közelítése T 0 T G(u)dB(u) = ( 1) α 0 +G(0+)B(T ), D α 0+G 0+ (u)d 1 α T B T (u)du ahol G 0+ I α 0+ (L 2(0, T )), B T I 1 α T (L 2(0, T ))
21 Tétel α (1 H, 1). G teljesíti: (1) G 0+ I0+ α (L 2(0, T )); (2) (3) T 0 T 0 ekkor D α 0+G 0+ (u) du < ; D0+ α G 0+(u) du <. (T u) 1 α t lim N 0 G(u)dB N (u) = t 0 G(u)dB(u) t [0, T ].
22 Tétel B fbm, B N közelítése, (C1) és (C2) teljesül. t 0 (0, T ] Ekkor t t X(t) = X 0 + F(X(s), s)ds + t 0 t 0 G(X(s), s)db(s), t X N (t) = X 0 + t 0 t F(X N (s), s)ds+ t 0 G(X N (s), s)db N (s), N N létezik mindkét egyenletnek megoldása (t 1, t 2 )-n és P( lim sup N t (t 1,t 2 ) X N (t) X(t) = 0) = 1.
23 Lépcsős fbm - mérnöki - biofizika - gazdasági modellezésben H : R ]0, 1[ K H(t) = a i 1 [τi,τ i+1 [(t), i=0 ahol τ o =, τ K +1 =,, τ 1, τ 2,...τ K növekvő és a i [0, 1].
24 Wavelet közelítés (B(t)) t [0,1] -re {2 j/2 Ψ(2 j x k) : (j, k) Z 2 } Lamarie Meyer wavelet L 2 (R)-ben, Ψ: Ψ 1 ixy Ψ(y) Ψ 1 (x, θ) = e dy, R y θ+ 1 2 Ayache,A.,Bertrand,P.R., Levy Vehel,J B(t)= 2 jh(k/2j ) (Ψ 1 (2 j t k,h(k/2 j )) Ψ 1 ( k, H(k/2 j )))ɛ j,k j= k= ahol ɛ j,k i.i.d. (4)
25 Ahol Ψ: Ψ C 1, c > 0 sup Ψ 1 (t, θ) θ [a,b] sup Ψ 1 (t, θ) θ [a,b] c (2 + t ) 2 (5) c, t R. (2 + t ) 3
26 -magas frekvenciájú komponens V 1 (t)= j=0 k= 2 jh(k/2j ) (Ψ 1 (2 j t k,h(k/2 j )) Ψ 1 ( k,h(k/2 j )))ɛ j,k alacsony frekvenciájú komponens V 2 (t)= 1 j= k= 2 jh(k/2j ) (Ψ 1 (2 j t k,h(k/2 j )) Ψ 1 ( k, H(k/2 j )))ɛ j,k. B(t) = V 1 (t) + V 2 (t) t [0, 1].
27 N N. B N 1 (t)= és B N 2 (t)= N j=0 2 k 2N+4 (N j+1) 2 1 j= 2 [N/2] jh(k/2 j ) (Ψ 1 (2 j t k,h(k/2 j )) Ψ 1 ( k,h(k/2 j )))ɛ j k 2 [N/2] 2 jh(k/2j ) (Ψ 1 (2 j t k,h(k/2 j )) Ψ 1 ( k, H(k/2 j )))ɛ B N (t) = B N 1 (t) + B N 2 (t) t [0, 1]. (6)
28 Figure: B N közelítése sfbm-nek
29 References Ayache,A.,Bertrand,P.R., Levy Vehel,J., A central limit theorem for generalized quadratic variation of the step fractional Brownian motion, SISP, 10, (2007), A. Ayache,A., Taqqu,M. S., Rate optimality of wavelet series approximations of fractional Brownian motion,jour. Fourier Anal. Appl. 9 (2003), Benassi,A.,Bertrand,P.R., Cohen,S. Istas,J., Identification of the Hurst index of a step fractional Brownian motion, Stat.Inference Stoch. Proc. 3, 1/2 (2000), Bender, C., Sottinen, T., Valkeila, E., Pricing by Hedging and no Arbitrage beyond Semimartingales, Finance Stoch. 12 (2008),
30 Meyer,Y., Sellan,F., Taqqu, M.R., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion The Journal of Fourier Analysis and Applications, 5 (1999), Lisei, H.,Soós, A, Approximation of Stochastic Differential Equations Driven by Fractional Brownian Motion, Seminar on stochastic analysis, random fields and applications V, Progress in Probability 59, Birkhäuser Verlag 2007,pp Robu, J., Soós, A, Approximation of Stochastic Differential Equations Driven by Step Fractional Brownian Motion, Annales Univ. Sci.Budapest, Annales Univ. Sci. Budapest., Sect. Comp. 37 (2012)
31 Samko,S.G., Kilbas,A.A., Marichev,O.I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach (1993). Zähle, M.,Integration with respect to Fractal Functions and Stochastic Calculus I., Probab.Theory Relat. Fields, 111 (1998), Zähle, M., Integration with respect to Fractal Functions and Stochastic Calculus II., Math. Nachr. 225 (2001),
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