Stoch. Integral & SDE (S. Hiraba) 1 1 (Definition of Stochastic Processes),, t, X t = X t (ω)., 1, 2,, n = 1, 2,..., X n = X n (ω).,., ω Ω,,.,,

Transcription

1 Stochastic Integrals and Stochastic Differential Equations (Seiji HIRABA) (Definition of Stochastic Processes) Brown (Wiener ) Poisson , C D (C Spaces and D Spaces) C D Skorohod Poisson (Stochastic Integrals) Wiener ( ) Poisson ( ) ( ) (Stochastic Differential Equations) (Transition Probabilities and Generators) ,,,,,,. (.)

2 Stoch. Integral & SDE (S. Hiraba) 1 1 (Definition of Stochastic Processes),, t, X t = X t (ω)., 1, 2,, n = 1, 2,..., X n = X n (ω).,., ω Ω,,.,, I R + = [, ),, (, Z +, N, N ), S (, d R d,.) (Ω, F, P ) {X t } t I, t I S X t = X t (ω). ( ω Ω,,.) I, (continuous time),, (discrete time)., X n, n =, 1, 2,...., S (state space).,,,, I = [, T ] or [, ), S = R 1. (filtration) (F t ) t, F σ-field. def {X t } (F t )- (adapted) t, X t F t,, X t F t. {X t }, (t, ω), i.e.,. (t, ω) ([, ) Ω, B 1 [, ) F) X t (ω) (R 1, B 1 ) {X t }, Ft, (Ft )-. = σ(x s ; s t) = s t X 1 s (B 1 ) = σ( s t X 1 s (B 1 )),, N = {N F; P (N) = }, i.e., F t = F t N = σ(f t N )., t, X t = Y t a.s., {Y t } (F t )-., {X t } (canonical filtration). 2 X = {X t }, Y = {Y t }, X Y def t, P (X t = Y t ) = 1 def X Y P ( t, X t = Y t ) = 1 X Y def t 1,, t n I, (X t1,..., X tn ) (d) = (Y t1,..., Y tn ) ( ),,., [ ],.,,, 1,,,.

3 Stoch. Integral & SDE (S. Hiraba) Brown (Wiener ) (B t ) t Brown (Brownian motion), (1) B = a.s. (2) (B t ), i.e., a.a.ω, B (ω). (3) = t < t 1 < < t n, {B tk B tk 1 } n,, N(, t k t k 1 ). 1, d Brown, B t = (B 1 t,..., B d t ) d Brown. (d Brown,.),, (3), B tk B tk 1 d N(, (t k t k 1 )I d ),. W = C([, ) R 1 ), σ- W. def, w = w(t) W w W ; w() =., t n = (t 1,..., t n ); t 1 < t 2 < < t n <, A n B n, C(t n, A n ) = {w W ; (w(t 1 ),..., w(t n )) A n } or (cylinder set). σ-, W. (, W σ-.) 1.1 (Wiener ) (Ω, F) = (W, W ),, B t (w) = w(t) Brown P B. P B Wiener.., Brown, Wiener, Brown Wiener (Wiener process)., d Brown B t = (B 1 t,..., B d t ), W d w; w C([, ) R d ), w() =, d Wiener.. p t (x) := 1 2πt d e x (x = (x 1,..., x d ) R d, x = x x2 d ), P (B t dx) = p t (x)dx. g t (x) d N d (, t)., (characteristic ft),. φ(z) = φ Bt (z) := E[e iz Bt ] = e t z 2 /2 (z R d )., z B t = z 1 B 1 t + + z d B d t., 1, p t (x, y) := p t (y x) = 1 2πt e (y x)2 /(2t), Brown < t 1 < t 2 < < t n A k B 1, P (B tk A k ) = dy 1 p t1 (, y 1 ) A 1 dy 2 p t2 t 1 (y 1, y 2 ) A 2 dy n p tn t n 1 (y n 1, y n ) A n

4 Stoch. Integral & SDE (S. Hiraba) 3.,, t =, P (B tk B tk 1 A k, k = 1, 2,..., n) = n A k p tk t k 1 (x k )dx k, x k = y k y k 1 (y = )., {B t1 A 1, B t2 A 2 } == {B t1 A 1, B t2 B t1 A 2 A 1 }. A 2 A 1,., (F t ) Brown (B t ). [Brown ] (1) EB 2n t = (2n 1)!!t n, EB 2n 1 t = (n 1). (2) s < t, B t B s F s.,.,, (B t ). i.e., s < t E[B t B s F s ] = (3) E[B t B s ] = t s (s, t > ). (4) (X t ) Brown def s < t, E[e iz(xt Xs) F s ] = e (t s)z2 /2., (F t ) (X t ). (5) Brown. (a > 1.), S a (B) t. B a t = B a+t B a, B t = B t, S a (B) t = ab t/a. (6) [T 1, T 2 ] Brown a.s., i.e., = {t k }; T 1 = t < t 1 < < t n = T 2, V = sup B tk B tk 1 = a.s. (7) ε >, (1/2 ε)-hödler,, γ >, lim sup h s t; t s h B t B s t s γ = or a.s. if γ < 1/2 or γ 1/2. (8) a.s. Brown. (9) (B t ) d Brown. T d, (T B t ) Brown., τ S := inf{t > ; B t S = Sr d 1 } S = B d (, r), B τs = B τs (ω)(ω) S.. (.) X t = tb 1/t Brown., X =. lim sup t B t 2t log log(1/t) = 1, lim inf t 1,, a.s. lim sup t B t 2t log log t = 1 a.s.

5 Stoch. Integral & SDE (S. Hiraba) 4 ε >, (1/2 ε)-hödler,. lim sup h s t; t s h B t B s 2 t s log(1/ t s ) = 1. [Brown ] (1), E[e izbt ] = e tz2 /2. (2) s 1 < < s n s < t Borel f(x, y), g(x 1,..., x n ), E[f(B t B s )g(b s1,..., B sn )] = E[f(B t B s )]E[g(B s1,..., B sn )]., n = 2, E[f(B t B s )g(b s1, B sn )] = f(x 4 x 3 )g(x 1, x 4 )p s1 (x 1 )p s2 s 1 (x 1, x 2 ) p s s2 (x 2, x 3 )p t s (x 3, x 4 )dx 1 dx 4 R 4 = f(y 2 )g(x 1, x 2 )p s1 (x 1 )p s2 s 1 (x 1, x 2 )p s s2 (y 1 )p t s (y 2 )dx 1 dx 2 dy 1 dy 2 R 4 = f(y 2 )p t s (y 2 )dy 2 g(x 1, x 2 )p s1 (x 1 )p s2 s 1 (x 1, x 2 )dx 1 dx 2 R 2 R = E[f(B t B s )]E[g(B s1, B s2 )], x 4 x 3 = y 2, x 3 x 2 = y 1, p s s2 (y 1 )dy 1 = 1., B t B s (B s1,..., B sn ),, Fs, N.,, E[B t B s F s ] = E[B t B s ] = (3) s < t, E[B t B s ] = E[(B t B s )B s + B 2 s] = EB 2 s = s. (4) ( ). ( ), E[e iz(xt Xs) F s ] = e (t s)z2 /2., s 1 < < s n < s < t Borel f(x 1,..., x n ), E[e iz(xt Xs) f(x s1,..., X sn )] = E[E[e iz(xt Xs) F s ]f(x s1,..., X sn )], X t X s F s, N(, t). = e (t s)z2 /2 E[f(X s1,..., X sn )]. (5) X t, E[e iz(xt Xs) F s ] = e (t s)z2 /2,,. (6) [T 1, T 2 ] = [, 1]. B t t [, 1] a.s.,, δ n = max 1 k n B k/n B (k 1)/n (n ) a.s., X n = n (B k/n B (k 1)/n ) 2, Z k = (B k/n B (k 1)/n ) 2 1/n, EZ 2 k = 3/n 2 2/n 2 + 1/n 2 = 2/n 2, E(X n 1) 2 =, {n k }; X nk 1 a.s., V = sup n EZk 2 = 2 n. B tk B tk 1 X n k δ nk a.s. (7) E B t B s 2n = c n t s n (c n = (2n 1)!!), Kolmogorov, γ < (n 1)/(2n) 1/2, γ-hölder.

6 Stoch. Integral & SDE (S. Hiraba) 5, γ = 1/2,, L 1, A n = { B k/n B (k 1)/n L/ n, k = 1, 2,..., n},, P ( B k/n B (k 1)/n L/ n) = P ( B k B k 1 L) = P ( B 1 L) =: p L (, 1),, P (A n ) = p n L., n 1 P (A n) <. Borel-Cantelli, P (lim sup A n ) =., 1, N = N(ω) 1; n N, k n, B k/n B (k 1)/n > L/ n. L 1,. (8) [, 1].. P ( s [, 1]; B s) P m 1, A m,n =, s [, 1]; B s, n+1 i+3 n N i=1 j=i+1 N 1 A m,n =. { B j/n B (j 1)/n 8m n }. m 1, t > s ; B t B s m(t s ), t [s, t ]., s k = ([ns] + k)/n, 1 k 4, s < s 1 s 4, s 4 s 4/n,, N 1; n N, s k [s, t ],, k = 2, 3, 4, B sk B sk 1 B sk B s + B s B sk 1 2m(s 4 s ) 8m n., m, N 1; n N, i = [ns ] + 1, 1 i n + 1, j = i + 1, i + 2, i + 3, B j/n B (j 1)/n 8m/n.,. m, N 1, P (A m,n ) =., ( ) Bj/n P B (j 1)/n 8m 2 8m/n = n 2π/n P (A m,n ) inf n N P n+1 n+1 lim inf n j+3 i=1 j=i+1 i+3 i=1 j=i+1 P e x2 /(2/n) dx = 2 8m/ n 2π { B j/n B (j 1)/n 8m n ( B j/n B (j 1)/n 8m n }. ) e x2 /2 dx C n. ( ) 3 C lim (n + 1) =. n n

7 Stoch. Integral & SDE (S. Hiraba) 6 (9),., z, x R d, z, T x = t T z, x, t < t 1 < < t n, z k R d (k = 1, 2,..., n), [ { }] n E exp i z k, T (B tk B tk 1 ) = e n (t k t k 1 ) t T z k 2 = e n (t k t k 1 ) z k 2, z k j, (T B t ) j Brown,,,., T, τ T S T B t S, τ T S = τ S, Brown, A B(S), P (B τs A) = P (T (B τ T S ) A) = P (T (B τs ) A) = P (B τs T 1 A). µ S (dξ) := P (B τs dξ) S., t, P (B t B(, r)) = p t (x)dx, P (τ S < ) = 1, µ S (S) = 1 B(,r). (, P (τ S = ) >, < P ( t >, B t B(, r)) lim sup t P (B t B(, r)) =.) [Brown ] 3,,. t [, 1]. [, T ],, [, ). D = n 1 {k/2n ; k =, 1,..., 2 n } [, 1] 2., R Kolmogorov, R D ( w = w(t) : D] R ), X t (w) = w(t) Brown P. (D, n,, Kolmogorov, D,.), Kolmogorov, (X t ) D a.s., X t = lim r t;r D X r, B t = X t. 1.2 (Kolmogorov ) (1) Banach (B, ) {X t } t D, C, α, β > ; E X t X s } α C t s 1+β, X t D a.s.,. (2) {X t } t [,1] s, t [, 1],, { X t } t [,T ], γ < β/α, γ-hölder. lim sup h s t; t s h X t X s γ t s = a.s. [ ] < γ < δ < β/α, δ n = 1/2 n. β αδ >. (1), E 2 n ( ) α Xkδn X (k 1)δn δ δ n 1 n n 1 2 n Cδ 1+(β αδ) n = C n 1 δ β αδ n <.

8 Stoch. Integral & SDE (S. Hiraba) 7, a.s., lim 2 n n ( ) α Xkδn X (k 1)δn = a.s. δ δ n, P ( n ; n n, k = 1, 2,..., 2 n, X kδn X (k 1)δn < δ δ n) = 1. 1 Ω, (1.1) n ; r, r D; < r r < δ n, X r X r C r r δ C = 2/(1 2 δ ), D., n n ; δ n+1 < r r δ n, r, r, [kδ n, (k + 1)δ n ], ((k 1)δ n, kδ n ), (kδ n, (k + 1)δ n ). r, r [kδ n, (k + 1)δ n ], X r X r X r X kδn + X kδn X r,, r kδ n = ε 1 δ n ε p δ n+p 2 ( p 1, ε i = or 1). r = kδ n, r j = kδ n + ε 1 δ n ε j δ n+j (j = 1, 2,..., p). r j r j 1 = ε j δ n+j δ n+j, Ω, p X r X kδn X rj X rj 1 < δn+j δ = δn+1/(1 δ 2 δ ). j=1 j=1 X kδn X r,,. (r r > δ n+1.) X r X r 2δn+1/(1 δ 2 δ ) 2(1 2 δ ) 1 r r δ. r ((k 1)δ n, kδ n ), r (kδ n, (k + 1)δ n ), kδ n, kδ n r, r kδ n < δ n, 2,., (1.1). (2) (1), X t D, X t t D X t, t / D, X t = lim r D;r t,, n ; s, t [, 1]; < t s < 1 2 n, X t X s C t s δ. γ < δ < β/α, γ-hölder., t / D, r n D; r n t, E X t X t } α lim inf n E X rn X t α C lim inf n r n t 1+β =, t [, 1], P (X t = X t ) = 1., (X t ) ( X t ).,. 1.3 Poisson α >, τ = τ(ω) α P (τ > t) = t αe αs ds = e αt

9 Stoch. Integral & SDE (S. Hiraba) 8. τ f(s) = αe αs. τ α- or (exponential time).,. E[τ] = αse αs ds = 1 α, V (τ) = E[τ 2 ] (E[τ]) 2 = 1 α τ, (memoryless property). t, s, P (τ > t + s τ > s) = P (τ > t). P (τ > t + s τ > s) = P (τ > t + s) P (τ > s) = e (t+s) e s = e t = P (τ > t). 1.2 τ 1, τ 2,... τ n, α 1, α 2,..., α n, min{τ 1, τ 2,... τ n } α 1 + α α n -. P (min{τ 1, τ 2,... τ n } = τ k ) = n = 2, k = 1. α k α 1 + α α n. P (τ 1 τ 2 } > t) = P (τ 1 > t, τ 2 > t) = P (τ 1 > t)p (τ 2 > t) = e (α1+α2)t. τ 1, τ 2,,. P (min{τ 1, τ 2 } = τ 1 ) = P (τ 1 < τ 2 ) = = = dsα 1 e α1s P (s < τ 2 ) dsα 1 e α1s e α2s α 1 α 1 + α A B, A 1-, B 2-.,,.. 3-, 1/3. λ >, (X t ) t λ Poisson ( ) ( λ-poisson ).

10 Stoch. Integral & SDE (S. Hiraba) 9 (1) X =, (2) s < t X t X s λ(t s) Poisson., (3) X t. P (X t X s = k) = e λ(t s) λk (t s) k k! (k =, 1, 2,... )., < t 1 < t 2 < < t n, X t1, X t2 X t1,..., X tn X tn (Poisson ) σ 1, σ 2,..., λ-. τ n = n σ k, τ =, X t = n τ n t < τ n+1, X t := n1 [τn,τ n+1)(t) = max{n; τ n t}, λ-poisson.., (X t ) t λ-poisson, τ 1, τ 2,.... τ 1, τ 2 τ 1, τ 3 τ 2,..., λ-. i.e.,. 1.3 n λ- σ k τ = n σ k Γ(n, λ), P (τ < t) = (σ n ), P (σ σ n < t) = n= 1 (n 1)! λn s n 1 e λs ds. s 1+ s n<t λ n e λ(s1+ sn) ds 1 ds n u k = s 1 + s k (k = 1,..., n), s = u n, un λ n e λ(s1+ sn) ds 1 ds n = du n du n 1 s 1+ s n<t = = = un du n du n 1 u2 u3 1 du n (n 1)! un 1 n λ n e λun 1 ds (n 1)! λn s n 1 e λs du 1 λ n e λun du 2 u 2 λ n e λun 1.3 τ n σ n+1 Γ(n, λ) P (X t = n) = P (τ n t < τ n+1 = τ n + σ n+1 ) = = ds ds = e λt λ n (n 1)! 1 (n 1)! λn s n 1 e λs P (t < s + σ n+1 ) 1 (n 1)! λn s n 1 e λs e (t s)λ s n 1 ds = e λt λn t n. n!

11 Stoch. Integral & SDE (S. Hiraba) 1 P (τ n+1 > t + s, X t = n) = P (τ n+1 > t + s, τ n t < τ n+1 ) = P (τ n + σ n+1 > t + s, τ n t) = = du du 1 (n 1)! λn u n 1 e λu P (u + σ n+1 > t + s) 1 (n 1)! λn u n 1 e λu e λ(t+s u) = e λ(t+s) λn t n n! (1.2) P (τ n+1 > t + s X t = n) = e λs = P (τ 1 > s). m 1,. P (τ n+m > t + s X t = n) = P (τ m > s). m m + 1 m, P (τ n+m t + s < τ n+m+1 X t = n) = P (τ m s < τ m+1 ) = P (X s = m)., n, m 1, P (X t = n, X t+s X t = m) = P (X t = n, X t+s = n + m) n, = P (X t = n)p (X t+s = n + m X t = n) = P (X t = n)p (τ n+m t + s < τ n+m+1 X t = n) = P (X t = n)p (X s = m) P (X t+s X t = m) = P (X s = m) = e λ λm s m. m! m = P (X t+s X t = m) = e λs,.,, (1.2),, P (τ n > t + s X t = n) = P (τ n > t + s τ n t < τ n+1 ) = P (X t+s = n X t = n) = P (τ n t + s < τ n+1 X t = n) = e λs. P (X t = n, X t+s X t = ) = P (X t = n, X t+s = n) = P (X t = n)p (X t+s = n X t = n) = P (X t = n)e λs. n P (X t+s X t = ) = e λs., t 1 < < t k, P (X t = n, X t1 X t = n 1,..., X tk X tk 1 = n k ) = P (X t = n, X t1 = N + n 1,..., X tk = n + + n k ) = P (X t = n )P (X t1 t = n 1,..., X tk t = n n k )

12 Stoch. Integral & SDE (S. Hiraba) 11,. P (X t = n, X t1 X t = n 1,..., X tk X tk 1 = n k ) = P (X t = n )P (X t1 t = n 1 ) P (X tk t k 1 = n k ) = P (X t = n )P (X t1 X t = n 1 ) P (X tk X tk 1 = n k ) 1.4, (X t ): Markov (Markov process) def s < t Borel f, E[f(X t ) F s ] = E[f(X t ) X s ] a.s., ( )= E[f(X t s X = x] x=xs a.s., Markov (time-homogeneous MP). [Brown Markov ] Brown (B t ), (x + B t ) x Brown. (W = W d, W) P x, x + B t B t. B t (w) = w(t), w W., P x (B = x) = 1.,, N = {N W; x R d, P x (N) = }, F t = F t N, Ft = σ(b s ; s t)., Ft F t+ := ε> F t+ε. Ft = F t. s, W (shift operator) θ s θ s w(t) := w(t + s). 1.4 ((F t ) Markov ) Y W. x R d, s,. E x [Y θ s F s ] = E Bs [Y ] a.s. Borel f, s < t, Y = f(b t s ), Y θ s = f(b t ),, E x [f(b t ) F s ] = E Bs [f(b t s )] = E[f(B t s ) B = x] x=bs, Markov. [ ] f, F j R 1 Borel, Y = f(b t ), = j n f j(b tj ) t 1 < < t n, W- Y, 2,. Y = f(b t ), f(x) = e izx ( z R)., Y = e izbt, Y θ s = e izbt+s, E x [e izbt+s F s ] = E x [e iz(bt+s Bs) e izbs F s ] = E x [e izbt ]e izbs = E Bs [e izbt ] Y = j n f j(b tj ) t 1 < < t n,. n, n + 1

13 Stoch. Integral & SDE (S. Hiraba) 12. E x [Y θ s F s ] = E x E x f j (B tj+s) j n+1 F t 1+s F s n+1 = E x f 1 (B t1+s)e Bt1 +s f j (B tj t 1 ) j=2 F s n+1 = E Bs f 1 (B t1 )E Bt1 f j (B tj t 1 ) = E Bs f 1 (B t1 ) n+1 j=2 j=2 f j (B tj ) = E Bs [Y ]. 2, 3,,, Borel. n+1 n+1 f(x) = E x f j (B tj t 1 ) = E f j (x + B tj t 1 ). j=2 4, F t1. Markov, (F t ). 1.5 ((F t ) Markov ) W Y x R d, s, j=2 E x [Y θ s F s ] = E Bs [Y ] a.s., F t = F t. [ ] Markov Ft = F t. A, B Fs. Y = 1 B, Ỹ = E B s [1 B ], E x [1 B ; A] = E x [Ỹ ; A], i.e., E x[(1 B Ỹ ); A] =. Ỹ F s -, A Fs, 1 B = Ỹ a.s.,, B Fs ; 1 B 1 B a.s.,, B F s., Fs a st F s. Markov. Y = f(b t ); f Borel. ε >, A F s+ε, E x [f(b t+s+ε ) F s+ε ] = E Bs+ε [f(b t )] a.s., E x [f(b t+s+ε )1 A ] = E x [E Bs+ε [f(b t )1 A ]. ε, BM f,, E x [f(b t+s )1 A ] = E x [E Bs [f(b t )1 A ].,, Borel Y (Blumenthal -1 (zero-one law)) A F = F, P (A) = or [ ] A F, P x (A) = E x [1 A ] = E x [E x [1 A θ ]; A] = E x [E B [1 A ]; A] = E x [P x (A); A] = P x (A) 2. P x (A) = or 1., 1 Brown (B t ),,., τ (, ) := inf{t > ; B t > }, P (τ (, ) = ) = 1.

14 Stoch. Integral & SDE (S. Hiraba) 13,,. S (X t ) t (Continuous-time Markov Chain),. s, t, i, j, k ul S, u l < s (l l ), P (X t+s = j X s = i, X ul = k ul (l l )) = P (X t+s = j X s = i).,. P (X t+s = j X s = i) = P (X t = j X = i). q t (i, j) = P (X t = j X = i). 1.7 Poisson.,. 1.2 S,,. X t. t 1 < t 2 < < t n < t n+1, X t1, X t2 X t1,..., X tn+1 X tn,, X tn+1 X tn (X t1,..., X tn ), X tn+1 X tn X tn.. P (X tn+1 = j n+1 X tk = j k, k n) = P (X tn+1 X tn = j n+1 j n X tk = j k, k n) = P (X tn+1 X tn = j n+1 j n ) = P (X tn+1 X tn = j n+1 j n X tn = j n ) = P (X tn+1 = j n+1 X tn = j n ). (X t ): (martingale) E[X t F s ] = X s a.s. def s t, X t L 1,, martingale, (M t ), M t L 1, E[M t F s ] = M s a.s.., EM t = EM., E[X t F s ] X s a.s., (sub-martingale),,. EX EX s EX t.,, (super-martingale). 1.8 (Doob-Meyer ) (X t ): conti. sub-mart., (D), i.e., a >, {X τ a } τ (, τ.) X t = A t + M t ; (M t ): conti. mart., (A t ): conti., A =.,.

15 Stoch. Integral & SDE (S. Hiraba) 14 2 C D (C Spaces and D Spaces) (Polish sp.):,,,., R d, (, 1) R 1,. I, R I f : I R., C, D,. C = C(I) = I, D = D(I) = I 1., f : I R 1 1, I,. 2.1 C I C C = C(I), I, i.e., I = [a, b] ( < a < b < ),.. d u (f, g) = sup f(t) g(t). t I I, I n = [a n, b n ]; I = I n,.. d u (f, g) = 2 n (1 sup f(t) g(t) ). t I n 1. Weierstrass, n, n ), D Skorohod I 1 D D = D(I),,,. (f α = 1 [,α] I (α > ).), Billingsley ; Skorohod,. I, i.e., I = [a, b] ( < a < b < ), D D = D(I), Billingsley d B., Φ I,. φ Φ, λ(φ) = sup φ(t) φ(s) log s t t s, φ Ψ def λ(φ) <. d b (f, g) = inf φ Ψ { f φ g + λ(φ)}

16 Stoch. Integral & SDE (S. Hiraba) 15.,, d S ( Skorohod ), d S,,. d S (f, g) = inf φ Φ { f φ g + φ i }.,, 2 D,, (2 ),,. I, C. 2.3 (X t ) X a.s.,,,.,,,, (rcll=right-conti. has left-limit, or cádlág ( ) )., 1. T >, t [, T ),,,,,,.,,. (, t, t,, s < t. X t = X t = lim s t X s.,, t, t X t := X t X t,.),,. Brown, Markov,,,., Poisson, 1,, Poisson (Poisson random measure). 2.4 Poisson (Z, Z), λ(dz) σ. N(dz) = N(ω; dz) Z λ ( ) Poisson (Poisson random measure)1 def (1) a.a. ω Ω, N(ω; dz) (Z, Z). (2) A Z, λ(a) < N(A) λ(a) Poisson, i.e., λ(a) Poisson, λ(a) = N(A) = a.s. (3) A n Z:, N(A n ).

17 Stoch. Integral & SDE (S. Hiraba) 16, (2), N(dz) := E[N(dz)] = λ(dz)., Z,, Z = [, ) R m (t, z), Z = B 1 ([, ) R m ), ν(dz) R m, ν({}) =,, n 1, ν( z 1/n) <. (ν σ-.) ν Lévy., N(dtdz) dtν(dz)-poisson (τ k, ξ k ) N(dt, dz), i.e., N(dt, dz) = δ (τk,ξ k )(dt, dz)., k, j 1, P (τ k τ j ) = 1.,, 2., 1. P ( t, N({t} R m ) = or 1) = 1.,. ( [1].), R m = R 1., dtν(dz)-poisson T >, t [, T ]. Z = { z 1}, Z n = {1/(n + 1) z < 1/n} (n 1). n, ν n = ν Zn, [, T ] Z n λ n (dt, dz) λ(dt, dz) λ([, T ] Z n ) := dtν n(dz) T ν(z n ),, {Y n k (λ(dt, dz) := dtν(dz)) = (τ n k, ξn k )} k 1, i.e., P (Y n k dtdz) = λ n (dt, dz)., K n T ν(z n )-Poisson, {Y n k, K m; n, k 1, m }., K n K n N n (dt, dz) = δ Y n k (dt, dz) = 1 dtdz (Yk n ), N = N n n, N N n (A) = k K n = m k, m k Yk n A, m k ([, T ] Z n ) \ A N n (A) λ(a)-poisson, Poisson Poisson,. ( ) N (τ i, ξ i ), n, k 1; (τk n, ξn k ), n, m, k, j 1; (n, k) (m, j), P (τk n τ j m) = 1,i.e., P (τ k n = τ j m) =. M 1 1, [, T ] M, 1 l M, Z n,l = [(l 1)T/M, lt/m) Z n,, Z n,m = [(M 1)T/M, T ] Z n., λ n (Z n,l ) = 1/M., τ n k = τ m j, l; [(l 1)T/M, lt/m), Yk n, Yk n, Y j m, P (τ n k = τ m j ) P ( M l=1 { Y n k Z n,l, Y m j Z m,l } ) = (τ n k, ξn k ) Z n,l, Y m j = (τ m j, ξm j ) Z m,l M λ n (Z n,l )λ m (Z m,l ) = 1 M. l=1

18 Stoch. Integral & SDE (S. Hiraba) 17 3 (Stochastic Integrals) 3.1 Wiener ( ), (Ω, F, P ), (F t ) t,. (B t ) t (F t )-Brown (, BM.); B = a.s., t [, ),, t [, T ]., f(t) = f(t, ω) [, T ] Ω, (F t )-, dtp (dω) L 2 -, (Stochastic integrals, Ito integrals) f(r)db r =., f. f(r, ω)db r (ω) f(t) = f(t, ω) L 2 loc def f : [, ) Ω R 1 ;, (f(t)) t (F t )-, f(r) 2 dr < a.s. for t >.,, T >, f., f(t) L 2 T def f : [, T ] Ω R 1 ;, (f(t)) (F t )-, Ef(r) 2 dr < for t T. f(r, ω)db r (ω), BM, (, ω ), Riemann.,, dtp (dω),, L 2 -, f L 2 T., f(t, ω) = f a (ω)1 (a,b] (t); a < b; a, b [, T ], f a F a. f S, i.e. n f(t, ω) = f (ω)1 {} (t) + f tk 1 (ω)1 (tk 1,t k ](t)., = t t 1 < < t n = T, f tk 1, F tk 1 -., T. f 2 T := T Ef(t) 2 dt. 3.1 S: dense in L 2 T under T, i.e., f L 2 T, f n S; f f n T., (F t )-,., (progressively m ble) t >, (s, ω) ([, t] Ω, B[, t] F t ) f(s, ω) (R, B 1 )., L 2 T, (F t)-,. [ ], f L 2 T, f1 ( f m), f., ε >, f(t, ω) = 1 ε (t ε) f(r, ω)dr

19 Stoch. Integral & SDE (S. Hiraba) 18, f. (,, f (F t )-.), f L 2 T, f n (t, ω) = f(, ω)1 {} (t) + n f(t k 1, ω)1 (tk 1,t k ](t),, f n S,, f f n T. t k = k n T ( ) ( ) (F t )-adapted,., t >, f n (r, ω) = f(, ω)1 {} (r) + n f(t k, ω)1 (tk 1,t k ](r) (t k = k n t), B1 ([, t]) F t -,,. f(t) = f (ω)1 {} (t) + n f t k 1 1 (tk 1,t k ](t),. M t (f) f(r)db r := n f tk 1 (B tk t B tk 1 t) 3.1 [2], ; f(t) = n ft 1 k 1 [t k 1,t k )(t),., BM ( ), Poisson,. (F t)-, (F t)-adapted,(f t)-,,., lim sup ε,. t ε f(r, ω)dr. 3.2 {M t (f)}, EM t (f) =, EM t (f) 2 = E[M t (f) F s ] = M s (f) a.s., i.e., E ( ) 2 f(r)db r =, E f(r)db r = [ ] Ef(r) 2 dr, E f(r)db r F s = Ef(r) 2 dr., s < t, s f(r)db r a.s., M 2 c, M = a.s. L 2 -( )martingale, ; {M t (f)} M 2 c,,, M ( f), M(g) t = f(r)g(r)dr., M(f), M(g) t, L 2 -mart. (M t ), (N t ), Mt 2, (D), Doob-Meyer, A t ; A =, M 2 t A t. A t = M t, (M t ) 2 (quadratic variation)., M, N t := 1 4 ( M + N t M N t ) = 1 2 ( M + N t M t N t ),,, M t N t M, N t mart.. (M t ) (N t ) 2., ; t < t 1 < < t n = t,. M t = lim n (M tk M tk 1 ) 2 (in prob.)

20 Stoch. Integral & SDE (S. Hiraba) 19 Brown, B t = t. [ ] f f(t) = f a (ω)1 [a,b) (t), M(f) t = f a (B t b B t a ),, M(f) =. ; s < t, E[M(f) t F s ] = M(f) s a.s.. E[M(f) t M(f) s F s ] = E[f a (B t b B t a B s b + B s a ) F s ], s a, E[E[f a (B t b B t a ) F a ] F s ] = E[f a E[(B t b B t a ) F a ] F s ] =. s > a, E[f a (B t b B s b ) F s ] = f a E[B t b B s b F s ] =., E[M(f) t M(f) s F s ] = a.s. 2, [ E M(f) t M(g) t M(f) s M(g) s., M(f) t = s ] f(r)g(r)dr F s = f(r) 2 dr, M(f + g) t = (f + g) 2 (r)dr,. g(t) = g c 1 [c,d) (t), a c. s c, s > c., f L 2 T, f n, T, (M(f n ) t ) Cauchy, (L 2 T, T ), (M(f) t ),., f S., 3.1 f, g L 2 T, ( M(f) t = f(r)db r ) t M 2,c, M ( f), M(g) t = f(r)g(r)dr., f L 2 loc, { σ n = inf t > ; } f(r) 2 dr n, f n (t) := f(t σ n ) L 2 T, M(f) t = lim M(f( σ n )) t. ( ω, N = N(ω) 1; n N, σ n (ω) = T.), M(f) t σn = M(f( σ n )) t., (M(f) t ) M 2,loc,c ; L2 -martingale. 3.2 Poisson ν(dz) R m Lévy,,. ν({}) =, n 1, ν( z 1/n) < N(dt, dz) dtdν-poisson,, N(dt, dz) := E[N(dt, dz)] = dtν(dz) [, ) R m Poisson. (F t ), s < t, U B m, N((s, t] U) F s. Ñ := N N Poisson (compensated Poisson random measure)., (τ k, ξ k ) N(dt, dz) N(dt, dz) = δ (τk,ξ k )(dt, dz).

21 Stoch. Integral & SDE (S. Hiraba) 2, τ k a.s.,., ξ k 1/n k a.s.. f(t, z) = f(t, z, ω), g(t, z) = g(t, z, ω) R d, (t, z, ω) [, ) R m Ω, (F t )-., P (1) (2) h(t, z, ω) σ-filed, f, g P-. f, g (F t )- (predictable). (1) (z, ω), t h(t, z, ω). (2) t >, (z, ω) h(t, z, ω) B m F t -, ( ) Poisson, N((s, t] U) F s,.,,,.,. t >, σ n : ST ; a.s., t >, n 1, dr f(r, z) ν(dz) < R m a.s., dr E g(r σ n, z) 2 ν(dz) <. R m,, + t+, (, t]., = + + = (,t].,, X t = f(r, z)n(dr, dz) := f(τ k, ξ k ).. s < t,, g,,, Y t = ( z 1) E[X t F s ] = X s + E X t t >, s ( z 1) k;τ k t E[f(r, z) F s ]drν(dz). dr E f(r, z) ν(dz) <. R 1 dr E g(r, z) 2 ν(dz) <. R m R m g(r, z)ñ(dr, dz) := L2 - lim n ( z 1/n) g(r, z)ñ(dr, dz).., Y t L 2 -martingale i.e., [ ] s E g(r, z)ñ(dr, dz) F s = g(r, z)ñ(dr, dz). R m R m, [ ( ) 2 ] E g(r, z)ñ(dr, dz) = R m dr Eg(r, z) 2 ν(dz). R m g, Y t = g(r, z)ñ(dr, dz) = lim R 1 n g(r σ n, z)ñ(dr, dz). R 1. L 2 -martingale,, Y t σn L 2 -martingale,.

22 Stoch. Integral & SDE (S. Hiraba) 21,, m = 1, T >, [, T ],, f(r, z),. f(r, z) = 2 n f(r n k 1)1 (r n k 1,r n k ] (r)1 U (z) (r n k = kt/2n, f(t) = f(ω; t) (F t )-, U B 1 ; ν(u) <.) 3.3 F, f(t, z, ω), 2,. (1) F f(t, z, ω) ; f in t, B m F t - in (z, ω). (2) f n F; f = f F (.) g(r, z),. [ ] E g(r, z)ñ(dr, dz) F s = a.s. s R. (s, t] 2 n {r n k },., Z n = { z 1/n}, U Z n = B m Z n.. s r n k 1 < rn k t. [ ] E g(rk 1)1 n (r n k 1,rk n] (r)1 U (z)ñ(dr, dz) F s s R [ [ ] ] = E E g(r k 1)Ñ((rn n k 1, rk n ] U) Fr n Fs k 1 [ ] ] = E g(rk 1)E [Ñ((r n n k 1, rk n ] U) Fs = a.s. 2, N. ] ] n E [Ñ((rk 1, rk n ] U) Fr n n k 1 = E [Ñ((rk 1, rk n ] U) =, g n (r, z) = g(r, z)1 ( z 1/n) L 2 -,., [ ] E g n (r, z)ñ(dr, dz) F s = a.s. s, g, A F s, [ ] E g(r, z)ñ(dr, dz); A =, 2. ( E [ = s R s R s R ] [ g(r, z)ñ(dr, dz); A E dr R E[(g n (r, z) g(r, z)) 2 1 A ]ν(dz). s R g n (r, z)ñ(dr, dz); A ]) 2 2,,, j k, g(r n k 1)Ñ((rn k 1, r n k ] U)g(r n j 1)Ñ((rn j 1, r n j ] U)

23 Stoch. Integral & SDE (S. Hiraba) 22, j < k, i.e., j k 1 g(rj 1 n )Ñ((rn j 1, rn j ] U)g(rn k 1 ) Ñ((rn k 1, rn k ] U),. j = k,, E[g(r n k 1) 2 Ñ((r n k 1, r n k ] U) 2 ] = E[g(r n k 1) 2 ]E[Ñ((rn k 1, r n k ] U) 2 ], E[Ñ((rn k 1, r n k ] U) 2 ] = EN((r n k 1, r n k ] U) = (r n k r n k 1)ν(U),, L 2 -. [ 3.3 ], z R m. (, F, B m F t F t.), D 2 [, ) Ω Dynkin (d-system) (i) [, ) Ω D. (ii) A, B D; A B B \ A D. (iii) A n D A n D. def, C 2 [, ) Ω d-system, d(c) C 2 [, ) Ω,, d(c) = σ(c).. (.) 3.3.,, A P, 1 A F., A P def 1 A F, Dynkin., k n, (Yt k ) (F t )-adapted, B k B 1, B k } C, C P, d(c) P, k n {Y k t, d(c) = σ(c) = P, P P. C P, n = 1, i.e., A t = {Y t B} P, φ n : on R 1, φ n 1 B, 1 At = 1 B (Y t ) = lim φ n (Y t ), F (.) σ- Dynkin, d(c) σ(c)., d(c), σ-. A, B d(c) A B d(c). A d(c), D A = {B [, ) Ω; A B d(c)}.., A d(c), D A Dynkin, d(c) D A,., B 1, B 2 D; B 1 B 2, A (B 2 \B 1) = (A B 2)\(A B 1) d(c). B n D A;, A ( B n) = (A B n) d(c).,, A C D A d(c),, B d(c), B D A, A B d(c), A B, A d(c) D A d(c) ( ) (X t ) R d, x,. X t (ω) = x + a(r, ω)dr + b(r, ω)db r (ω).

24 Stoch. Integral & SDE (S. Hiraba) 23, X t = (X i t) = (X 1 t,..., X d t ); X i t = x i +, B t = (B k t ) k N : N Brown, a i (r)dr + a(t) = a(t, ω) = (a i (t, ω)) i d : (F t )-adapted; T >, b i k(r)dbr k. j N T b(t) = b(t, ω) = (b i k (t, ω)) i d,k N : (F t )-adapted; T >, {b(t)} t L 2,loc., a(t) dt < a.s. T b(t) 2 dt < a.s.,, b(t)db t = k N b i k(t)db k t, b(t) 2 = i,k (b i k) 2 (t).,. (,.) dx t = a(t)dt + b(t)db t., (Ito s formula), : φ(x) C 2 (R d ), dφ(x t ) = a(t) Dφ(X t )dt + b(t) Dφ(X t )db t b2 (t) D 2 φ(x t )dt., a(t) D = a i (t) i, b 2 (t) D 2 = k m bi k (t)bj k 2 ij ). i = / x i, ij 2 = 2 / x i x j. d = 1, N = 1, (, dφ(x t ) = φ (X t )a(t)dt + φ (X t )b(t)db t φ (X t )b(t) 2 dt., f Taylor,, (db t ) 2 = dt, (dt) 2 = dtdb t =,., n 3, (dx t ) n =,. dφ(x t ) = φ (X t )dx t + φ (X t )dx t φ (X t )(dx t ) 2 = φ (X t )dx t + φ (X t )dx t φ (X t )(dx t ) 2 = φ (X t )(a(t)dt + b(t)db t ) φ (X t )(b(t)db t ) 2. [ ] a(r), b(r),, [s, t],, r [s, t], a.s. a(s), b(s) F s., [s, t] n,, s =, a, b F, a.s., [, t] n, t k = t n k = tk/n, k =, 1,..., n. Taylor, φ(x tk ) φ(x tk 1 ) = φ (X tk 1 )(X tk X tk 1 ) φ (Y k )(X tk X tk 1 ) 2

25 Stoch. Integral & SDE (S. Hiraba) 24, θ = θ(ω) (, 1); Y k = X tk 1 + θ(x tk X tk 1 ). X tk X tk 1 = a (t k t k 1 ) + b (B tk B tk 1 )., n, a.s. in L 2,., X tk X tk 1 a(t)dt + b(t)db t (X tk X tk 1 ) 2 (a(t)dt + b(t)db t ) 2 = b(t) 2 dt φ(x t ) φ(x ) = n (φ(x tk ) φ(x tk 1 )) =, φ Cb 2 (φ ), n,. n In 1 = φ (X tk 1 )a (t k t k 1 ) φ (X r )a(r)dr a.s., I 2 n = I 3 n = I 4 n = I 5 n = n φ (X tk 1 )b (B tk B tk 1 ) n n n 1 2 φ (Y k )b 2 (B tk B tk 1 ) j=1 I j n φ (X r )b(r)db r in L 2, φ (X r )b 2 (r)dr in L 2, 1 2 φ (Y k )a b (t k t k 1 )(B tk B tk 1 ) a.s., 1 2 φ (Y k )a 2 (t k t k 1 ) 2 a.s., a.s.,. I 4 n, B r max B tk B tk 1 a.s., I 5 n, t k t k 1 = 1/n. I 1 n, I 2 n, I 3 n. I1 n, φ (X r ) r, In 1 a φ (X r )dr = φ (X r )a(r)dr In 2 f n(r) = n φ (X tk 1 )1 (tk 1,t k ](r), In 2 = b f n (r)db r, φ, φ (X r ), f n (r) φ (X r )1 (,t] (r) T., In 2 = b f n (r)db r b φ (X r )db r = In 3, ψ = φ C b (R), n ψ(y k )(B tk B tk 1 ) 2 p.w. φ (X r )b(r)db r in L 2. ψ(x r )dr in L 2, X r ψ, max 1 k n E ψ(y k ) ψ(x tk 1 ) 2, ψ ( n 2 n E {(B tk B tk 1 ) 2 (t k t k 1 )}) = E{(B tk B tk 1 ) 2 (t k t k 1 )} 2 = t2 n,., φ C 2, σ n = inf{t > ; X t n}, X t σn, { x n} φ, φ, Ito, n.

26 Stoch. Integral & SDE (S. Hiraba) ( ) ν(dz) R m Lévy,,. ν({}) =, n 1, ν( z 1/n) < N(dt, dz) dtdν-poisson,, N(dt, dz) := E[N(dt, dz)] = dtν(dz) [, ) R m Poisson. Ñ := N N Poisson. f(t, z) = f(t, z, ω), g(t, z) = g(t, z, ω) R d (F t )-,,. t >, dr f(r, z) ν(dz) <, a.s., ( z 1) σ n : ST; a.s., t >, n 1, dr E g(r σ n, z) 2 ν(dz) <. ( z <1) (X t ) x,. a, b, dx t (ω) = a(t, ω)dt + b(t, ω)db t (ω) + f(t, z, ω)n(ω; dt, dz) + g(t, z, ω)ñ(ω; dt, dz), ( z 1) ( z <1) : φ(x) C 2 (R d ), dφ(x t ) = a(t) Dφ(X t )dt + b(t) Dφ(X t )db t b2 (t) D 2 φ(x t )dt + [φ(x t + f(t, z)) φ(x t )]N(dt, dz) + + ( z 1) ( z <1) ( z <1) [φ(x t + g(t, z)) φ(x t )]Ñ(dt, dz) [φ(x t + g(t, z)) φ(x t ) g(t, z) Dφ(X t )]ν(dz)dt : Ikeda-Watanabe [4], z 1, z < 1, fg =, f, g z R.,, i.e., a =, b =,. dx t = f(t, z)n(dt, dz) + g(t, z)ñ(dt, dz), dφ(x t ) = ( z 1) ( z 1) + + ( z <1) ( z <1) ( z <1) [φ(x t + f(t, z)) φ(x t )]N(dt, dz) [φ(x t + g(t, z)) φ(x t )]Ñ(dt, dz) [φ(x t + g(t, z)) φ(x t ) g(t, z) Dφ (X t )]ν(dz)dt., d = m = 1, a =, b =.,,.,. dr E f(r, z) ν(dz) <, ( z 1) dr E g(r, z) 2 ν(dz) <. ( z <1)

27 Stoch. Integral & SDE (S. Hiraba) 26, g =, X t pure jump, φ(x t ) f(r, z) jump, φ(x t ) φ(x ) = = r t; X r ( x 1) (φ(x r ) φ(x r )) = r t; X r (φ(x r + f(r, z)) φ(x r ))N(dr, dz). (φ(x r + f(r, z)) φ(x r )) g, f =,,. dxt n := g(t, z)ñ(dt, dz), dxt n := (1/n z <1) (1/n z <1) g(t, z)n(dt, dz) g(t, z)dtν(dz) (1/n z <1)., dxt n = d(xt n ) d + d(xt n ) c = Xt n + d(xt n ) c. Xt n = Xt n Xt, n i.e, Xt n = Xt + X n t n.,, N({(t, z)}) = 1, Xt n = g(t, z), φ(x n t ) = φ(x n t ) φ(x n t ) = φ(x n t + g(t, z)) φ(x n t )., Z n = {1/n z < 1}. φ(xt n ) φ(xs n ) = φ (Xr n )d(xr n ) c + φ(xr n ) s r; Xr n t = φ (Xr n )g(r, z)drν(dz) + [φ(xr n + g(r, z)) φ(xr )]N(dt, n dz) s Z n s Z n = φ (Xr n )g(r, z)drν(dz) + [φ(x n n r + g(r, z)) φ(xr )]Ñ(dr, dz) s Z n s Z n + [φ(xr n + g(r, z)) φ(xr )]drν(dz) n s Z n = [φ(x n n r + g(r, z)) φ(xr )]Ñ(dr, dz) s Z n + [φ(xr n + g(r, z)) φ(xr ) n φ (Xr n )g(r, z)]drν(dz) s Z n, n, L 2 -,.

28 Stoch. Integral & SDE (S. Hiraba) 27 4 (Stochastic Differential Equations), 1 (d = m = 1). a(t, x), b(t, x), f(t, x, z), g(t, x, z); g(t, x, ) = R 1,,, (t, x.z) [.T ] R 1 R 1. (, ω.), R 1 X t. X = x, dx t = a(t, X t )dt + b(t, X t )db t + ( z 1) f(t, X t, z)n(dt, dz) + g(t, X t, z)ñ(dt, dz). ( z <1), X = x, a, b, f, g. (Lévy ν, a, b, f, g, ν.),, ( ), a, b Brown, f, g. (,, a.s..) f, g, i.e, f = g =, (continuous type),, = (jump type)., a = b =, (pure jump type).,,,,,,,,,. X t = x + a(s, X s )ds + b(s, X s )db s + + ( z 1) ( z <1) f(s, X s, z)n(ds, dz) g(s, X s, z)ñ(ds, dz). 4.1 d 1, N 1. B t = (Bt) i i N : N Brown ; B =. (t, x) [, T ] R d Borel a, b, a = a(t, x) = (a i (t, x)) i d : d, b = b(t, x) = (b i k (t, x)) i d,k N: d N. X = x R d. dx t = a(t, X t )dt + b(t, X t )db t. N dxt i = a i (t, X t )dt + b i k(t, X t )dbt k, 1 i d. a ( ), b. (,,,.), X i t = x i + X t = x + a(r, X r )dr + a i (r, X r )dr + N b(r, X r )db r. b i k(r, X r )db k r, 1 i d.

29 Stoch. Integral & SDE (S. Hiraba) 28 a 2 = (a 1 ) (a d ) 2, b 2 = i d,,k N (bi k )2. 4.1,, x, 1,, Lipschitz, i.e., T >, K = K T > ; t [, T ], x R d, a(t, x) + b(t, x) K(1 + x ), a(t, x) a(t, y) + b(t, x) b(t, y) K x y, (X t ) t,,, (X t ) L 2., T >, T E X t 2 dt <., ( X t ),, i.e., P (X t = X t, t ) = 1. [ ] Picard. X n = (X n t ). X 1 t = x, X n, dx n+1 t = a(t, X n t )dt + b(t, X n t )db t, X n+1 = x.,. X n = (Xt n ) (F t )-adapted, [ ] E sup Xt n 2 <. t T X n+1 well-defined,. T >, n 1, [ ] E sup Xt n+1 Xt n 2 (C 1 t) n 1 C 2 t T (n 1)!, {X n } a.s., X = (X t ), T, SDE., X, X 2, τ L = inf{t ; X t X t L},, E X t τl X t τl 2 C 1 E X r τl X r τl 2 dr. Gronwall ( ), E X t τl X t τl 2 =, t T., X, X, P ( t T, X t τl = X t τl ) = 1 τ L a.s.,,. 4.1 (Gronwall ) [, T ] g(t), C 1, C 2 ; g(t) C 1 + C 2 g(r)dr g(t) C 1 e C2t [ ] h(t) = e C2t C 2g(r)dr,, h (t) C 1 C 2 e C2t. h() =, [, t], h(t) C 1 (1 e C2t ).,. 4.2,, (t, x, z) [, ) R d R m Borel, f = f(t, x, z) = (f i (t, x, z)) i d ; f = on z < 1, g = g(t, x, z) = (g i (t, x, z)) i d ; g(t, x, ) = g = on z 1. dx t = a(t, X t )dt + b(t, X t )db t + f(t, X t, z)n(dt, dz). + ( z 1) ( z <1) g(t, X t, z)ñ(dt, dz).

30 Stoch. Integral & SDE (S. Hiraba) 29 ν(dz) R m, n 1, ν( z 1/n) <. N(dt, dz) dtν(dz)-poisson r.m., N(dt, dz) = dtν(dz), Ñ = N N. f. t >, x R d, dr f(r, x, z) ν(dz) <. ( z 1) 4.2,, T >, K = K T > ; t [, T ], x R d, g(t, x, z) 2 ν(dz) K(1 + x 2 ), ( z <1) g(t, x, z) g(t, y, z) 2 ν(dz) K x y 2 ( z <1), (X t ) t, 1., f =, (X t ) L 2. [ ],, T >, N([, T ] { z 1}) < a.s., T =, (τ k, ξ k ); < τ k., f = Yt 1,., Xt 1 = Yt 1 for t [, τ 1 ), Xt 1 = Yτ f(τ 1, Y (τ 1 ), ξ 1 ) if t = τ 1 ),. Yt 2 t [, τ 2 τ 1 ], Y 2 = Xτ 1 1 Bt τ1 := B t+τ1 B τ1, N τ1 (dt, dz) := N(dt + τ 1, dz) Xt 1. (, r r + τ 1.) Xt 2 = Xt 1 if t [.τ 1 ], Xt 2 = Yt τ 2 1 if t [τ 1, τ 2 ], [, τ 2 ]., Yt 2,, τ 1, s, x R d, Yt 2 = Yt 2 (ω; x, s), s = τ 1 (ω), x = Xτ 1 (ω).,, X 1(ω) t, X t = Xt k for t [, τ k ],. [ ] [Xt 2 t [, τ 2], t (τ 1, τ 2 ] ] s = τ 1, t [, τ 2 s), Y 2 t X 1 s = = +s s a(r + s, Y 2 r )dr + a(r, Y 2 r s)dr + +s s b(r + s, Y 2 r )db s r + b(r, Y 2 r s)db r + ( z <1) +s s ( z <1), s, (B s r = B r+s B s ) (d) = (B r )., b(r + s, Y 2 r )db s r = b(r + s, Y 2 r )d(b r+s B s ) = +s s g(r + s, Y 2 r, z)ñ s (dr, dz) g(r, Y(r s) 2, z)ñ(dr, dz) b(r, Y 2 r s)db r. t [s, τ 2 ), Xt 2 = Yt s 2 Y t 2,.,. X 2 τ 2 = Y 2 τ 2 s + f(τ 2, Y (τ2 s), ξ 2 )

31 Stoch. Integral & SDE (S. Hiraba) 3 5 (Transition Probabilities and Generators) (X t, P x ) R d x. Borel φ, P t (x, dy) := P x (X t dy), P t φ(x) := E x [φ(x t )] = φ(y)p t (x, dy) R d. P t (x, dy),, (P t ) t C b C b (R d ) (transition semi-group)., P = I, P s P t = P s+t (s, t ). (X t ) D D(R d ) Markov, lim t P t φ(x) = φ(x), i.e., lim t P t I on C b., P t + h P t (h ) on C b for t. (P t ) on C b, (X t ), i.e., ε >, P ( X t+h X t < ε) 1 (h ) for t., φ C b φ() =, φ(x) = 1 if x ε, P ( X t+h X t ε) E[φ(X t+h X t )] = E[ φ(y x)p h (x, dy) x=xt ] E[φ()] =. 5.1, (P t ) C b. L on D(L) C b : 1 L := lim h h (P 1 h I), i.e., Lφ(x) := lim h h (P hφ(x) φ(x))., φ D(L) φ C b., P t = e tl. s t, P s,t (x, dy) = P (X t dy X s = x),, P s,s = I, P s,t P t,u = P s,u.,.,, L t = lim h (P t,t+h I)/h on D(L t )., P s,t = exp[ s L rdr].. φ Cb 2 C2 b (Rd def ) ij 2 φ C b (i, j d). ij 2 = 2 / x i x j. φ Cc def φ C, supp φ: compact. 5.1 (X t ) X = x, i.e., (a, b, f, g) 4.2, dx t = a(t, X t )dt + b(t, X t )db t + f(t, X t, z)n(dt, dz). + ( z 1) ( z <1) g(t, X t, z)ñ(dt, dz).

32 Stoch. Integral & SDE (S. Hiraba) 31, φ C 2 b,. L t φ(x) = a i (t, x) i φ(x) + 1 N b i 2 kb j k (t, x) 2 ijφ(x) + [φ(x + f(t, x, z)) φ(x)]ν(dz) + ( z 1) ( z <1) [φ(x + g(t, x, z)) φ(x) i φ(x)g i (t, x, z)]ν(dz).,, e.g. a i x i = i d ai x i. [ ] d = 1,, Ito φ(x t+h ), φ(x t+h ) φ(x t ) = h t +h t +h t ( z 1) ( z <1) ( z <1) +h t φ (X s )d(x s ) c h t φ (X s )d(x 2 s ) c [φ(x s + f(s, X s, z)) φ(x s )] N(ds, dz) [φ(x s + g(s, X s, z)) φ(x s )] Ñ(ds, dz). [φ(x s + g(s, X s, z)) φ(x s ) φ (X s )g(s, X s, z)] dsν(dz)., d(x s ) c = a(s, X s )ds + b(s, X s )db s, d(x 2 s ) c = (db s ) 2 = ds. P t,t+h φ(x) = E[φ(X t+h ) X t = x],, P t,t+h φ(x) φ(x) = E[φ(X t+h ) φ(x t ) X t = x], db s Ñ,., Brown, X t = x + B t,, L t L = x /2.,, Markov, Lévy, f = g = z (m = d), ν. ν({}) =, (1 z 2 )ν(dz) <. R d Lφ = a Dφ b2 D 2 φ + [φ( + z) φ( ) z Dφ( )1 ( z <1) ]ν(dz) R d., a D = a i i, b 2 D 2 = k bi k bj k 2 ij., E[e iξ Xt ] = e tψ(ξ) ; Ψ(ξ) = ia ξ 1 2 b2 ξ 2 + [e iz ξ 1 iz ξ1 ( z <1) ] ν(dz)., ξ R d, b 2 ξ 2 = k bi k bj k ξ iξ j.

33 Stoch. Integral & SDE (S. Hiraba) (string solution).,,,., (a, b, f, g, ν),,, (weak solution)., L t, P s,t, ( D(L t )., L t a, b, f, g Lévy ν(dx), (5.1 )., L t (5.1). D(L t )., t> D(L t).,,,.,,, well-posed ( ). Ω = D([, )), F = σ(c); C cylinder set, i.e., C C def C = {(ω(t 1 ),..., ω(t n )) A t1 A tn, ( t 1 < < t n, A tj B 1, n 1), F t t cylinder sets σ-, F t := ε> F t+ε., x R d, (Ω, F, (F t )), P = P x, X t (ω) = ω(t), {P x } x. D (C c Cb 2 ), (1) P (X ) = x (2) φ D, M(φ) := φ(x t ) φ(x), Markov, L t. L r φ(x r )dr (F t )-martingale.,.,, D D(L t ),, (L t ) (core)., P x. s < t, P s,t φ(y) = E x [φ(x t ) X s = y], P s,t φ(y) φ(y) = E [ φ(x t ) φ(x s ) X s = y] =., L t. s E x [L r φ(x r )] X s = y]dr =, t, t P s,t φ(x) = P s,t L t φ(x) = L t φ(y)p s,t (x, dy). s P s,r L r φ(y)dr d = 1., P t (x, dy) = p t (x, y)dy, y p, 2 yp t p y., L (f = g = ), a, b Cb 2, φ C2 c, ( ) φ(y) t p t (x, y)dy = φ(y)l p t (x, y)dy.

34 Stoch. Integral & SDE (S. Hiraba) 33, t p t (x, y) = L p t (x, y) a.e.., L L,. L ( p(y) = i a i (y)p(y) ) ij b i kb j k (y)p(y). k N, X t X, u(t, y) = E[p t (X, y)],, t u = L u a.e.. (,,.) [1], (1953). [2], (1997, 2 2). [3],,, [4] N. Ikeda, S. Watanabe Stochastic Differential Equations and Diffusion Processes, 2nd Ed., Holland-Kodansha (1988, 2nd.ed. 1991).