II Brown Brown

Transcription

1 II Brown Brown Markov Brown Martingales Stopping time Martingales Brown Markov

2 2

3 1 Brown 1 2 Brown Markov 1.1 (Ω, F, P t t [, t [, T, T > 1.1 (1 t R d X = (X t t = (X t (ω t d (stochastic process (d = (2 d ω Ω t X t (ω 1.1 W d = C([,, R d (path X d X(ω W d, ω Ω X W d - : A B(W d { ω Ω ; X(ω A} F W d 5 B(W d Borel σ- : B K (W d := σ { C(t 1,..., t n ; A ; n N, t 1 < < t n <, A B(R dn } C(t 1,..., t n ; A := {w = (w t t W d ; (w t1,..., w tn A } (cylinder set σ- ( σ- Kolmogorov σ- B(W d = B K (W d X W d (1 B(W d = B K (W d (2 X W d - F σ- (F t t F t σ- F s F t F, s t filtration 1 URL sugiura/ 2 [F [N : [IW N.Ikeda and S.Watanabe: Stochastic Differential Equations and Diffusion Processes. 2nd ed., North Holland, [KS I.Karatzas and S.E.Shreve: Brownian Motion and Stochastic Calculus. 2nd ed., Springer, [O B.Øksendal: Stochastic Differential Equations. An Introduction with Applications. 6th ed., Springer, 23. ( 5 : [RY D.Revuz and M.Yor: Continuous Martingales and Brownian Motion 3rd ed., Springer, [SV D.W.Stroock and S.R.S.Varadhan: Multidimensional Diffusion Processes, Springer, [F [F1 ( 3 path (Poisson 4 (continuous modification X Y P (X t = Y t = 1, t 5 W d ( 3

4 1.2 (1 d X = (X t t (F t - (adapted t X t : Ω R d F t - (2 d X = (X t t (F t - (progressively measurable t [, t Ω (s, ω X s (ω R d B([, t F t /B(R d - B([, t F t σ- 1.2 X (F t - t F t X t E[ f(x s ds = f(x Ω [,t s(ω P (dωds ( 1.1 X (F t - (F t X = (X t (F t - X : lim h X t+h (ω = X t (ω, t [,, ω Ω (F t - : X t X (n s (ω := X (k+1t/2 n(ω for kt/2 n < s (k + 1t/2 n, k =, 1,..., 2 n 1 A B(R d {(s, ω; X (n s (ω A} = 2 n 1 k= ( kt (k + 1t, 2n 2 n { ω ; X (k+1t/2 n(ω A } B([, t F t X s (n B([, t F t - lim n X s (n (ω = X s (ω, ω Ω. X s B([, t F t (F t t filtration (1 F t+ := s>t F s F t+ = F t, t, (F t (2 F t := σ(f s ; s < t F t = F t, t, (F t 1.1 X = (X t t t = σ{ X s ; s t } (F,X t t filtration X (F,X t - (F,X t X filtration (F,X t F,X 1.2 X d O R d σ O (ω := inf{ t ; X t (ω O } (inf = {σ O t} F,X t+ {σ O t} / F,X t P N := { N F ; P (N = } F,X t Ft X := F,X t N X t = Y t a.s. ( t X (F t X - Y (F t X - F t ( Brown t (cf

5 1.2 Brown 1.4 (Ω, F, P ( B = (B t t = (B t (ω t Brown (Brownian motion (1 B = a.s. (2 ω Ω t B t (ω (3 s t B t B s F s,b := σ(b u ; u s (4 s t B t B s, t s Gauss 1.3 (2 1.1 a.s.ω B t (ω ω B t (ω (t (3, (4 = t < t 1 < < t n {B ti B ti 1 } 1 i n N(, t i t i 1 A 1,..., A n B(R P (B ti B ti 1 A i, 1 i n = dx 1 A 1 dx 2 A 2 dx n A n n p(t i t i 1, x i p(t, x = 1 2πt e x2 /2t, t >, x R (1.1 y =, x i = y i y i 1, 1 i n A 1,..., A n B(R n P (B ti A i, 1 i n = dy 1 dy 2 dy n p(t i t i 1, y i y i 1 (1.2 A 1 A 2 A n Brown n 1.2 Brown 1.2 Kolmogorov (Kolmogorov n N (R n, B(R n µ n (consistent i=1 µ n+1 (A R = µ n (A, A B(R n, n N (R N, B(R N µ µ(a R N = µ n (A, A B(R n A R N := { (x 1, x 2,... ; (x 1,..., x n A } R N R N B(R N Borel 6 Brown (1 Fourier (2 Gauss (3 (Donsker (cf. [KS pp i=1

6 : 1 : C := { A R N ; A B(R n, n N } R N µ µ(a R N = µ n (A {µ n } µ n C C n C C n (C n C n+1, n N n=1c n = lim µ(c n = (1.3 n Hopf µ B(R N = σ(c (1.3 ε > : µ n (C n > ε, n N n=1c n 2 : C n = A n R N, A n B(R n C 1 = R R N C n A n A n A n A n µ n(a n \A n ε/2 n+1 i < n à i = A i R n i A n = à 1 à n 1 A n, C n = A n R N A n C n µ(c n = µ n (A n µ(a n n µ i (A i \A i ε 2 i=1 3 : n=1c n x n = (x 1 n, x 2 n,... C n, n N {x n } {x nk } x = (x 1, x 2,... R N i lim n x i n = x i A n (x 1,..., x n A n x n=1c n D n := { k/2 n ; k =, 1,..., 2 n }, D = n=1d n 1.4 (Kolmogorov X = (X t C, ε, p > E[ X t X s p C t s 1+ε, s, t D (1.4 X t a.s. D : 1 : λ n := 2 nε/2p I n := { (s, t D D ; t s 2 n } A n := { ω Ω ; X s X t λ n, (s, t I n } P (A n C 1 2 nε/2 Borel-Cantelli P ( n=1 k=n A k = 2 : a(n, t = k/2 n, t [k/2 n, (k + 1/2 n X t = X a(n,t + (X a(n+i+1,t X a(n+i,t, t D i= j a(j, t = t X s (s, t I n a(n, t a(n, s 2 n 6

7 3 : ω A n (a, (b : (a (s, t I n s.t. X a(n,t X a(n,s λ n /2 (b t, i s.t. X a(n+i+1,t X a(n+i,t 3λ n /2(i π 2. (a, (b X t X s < λ n 2 + 3λ n 1 π 2 i= = λ (1+i 2 n, (s, t I n (a (q, r I n (D n D n s.t. X r X q λ n /2 (q, r 2 n P ((a = 2 n sup P ( X s+2 n X s λ n /2 2 n (2/λ n p sup E[ X s+2 n X s p s D n s D n 2 n 2 p λ p n C2 n(1+ε = C 2 2 nε/2. (b i, (q, r I n+i (D n+i+1 D n+i s.t. X r X q 3λ n /2(i+1 2 π 2 (q, r 2 n+i+1 P ((b = 2 n+i+1 sup P ( X s+2 (n+i+1 X s 3λ n /2(i π 2 s D n+i i= C 2 n+i+1 2 (n+i+1(1+ε (2(i π 2 /3λ n p i= C 3 i= 2 (n+i+1ε (i + 1 2p λ p n C 4 2 nε/2. P (A n C 1 2 nε/2 1.5 X = (X t t [,1 (1.4 s, t [, 1 X : t / D X t := lim s D,s t X s 1.4 X t t [, 1 (1.4 s D, s t X s X t in L p t [, 1 X t = X t a.s. X t X t 1.2 : 1 : [, 1 Brown (B t t [,1 Ω = R D, B t (ω = ω t, t D, ω Ω D = {t 1, t } R n µ n {t 1,..., t n } s 1 < s 2 < < s n n µ n (dx 1 dx n := p(s i s i 1, x i x i 1 dx i, s =, x = i=1 p(t, x (1.1 (s 1 = p(, x 1 dx 1 = δ (dx 1 {µ n } p : p(t + s, y x = p(t, z xp(s, y z dz, t, s > R Chapman-Kolmogorov Kolmogorov Ω µ E µ [ B t B s 4 = x y 4 p(s, yp(t s, x y dydx = 3 t s 2, t > s, s, t D R 2 7

8 Kolmogorov (B t t D µ-a.s. t / D B t := lim s D,s t B s (B t t [,1 (Ω, µ Brown 2 : [, 1 Brown {(Bt i t [,1 } i N ( B t := [t i=1 Bi 1 + B [t+1 t [t, t (B t t Brown W (= W 1 := C([,, R, W := { w W ; w = } B(W, B(W Borel Ω := W, F = B(W B t (w = w t, w = (w t t W Brown (W, B(W P Wiener 1.2 Wiener 1.6 d B t = (Bt i 1 i d = (Bt 1,..., Bt d, t d Brown Brown {Bi } 1 i d B i t d Brown d Brown i = 1,..., d (Ω i, F i, P i Brown B i = (Bt i t Ω = d i=1 Ωi, F = d i=1 F i, P = d i=1 P i ω = (ω 1,..., ω d Ω B t (ω = (Bt 1 (ω 1,..., Bt d (ω d (Ω, F, P d Brown 1.3 B = (B t t (Ω, F, P 1 Brown Ft = σ{ B s ; s t }, F t = Ft N N = { N F ; P (N = } Brown 1.6 (1 E[B 2p t = (2p 1!!t p, E[B 2p 1 t =, p N. (2 E[B t B s = t s(:= min{t, s}, t, s. (3 B = (B t t (F t -martingale E[B t F s = B s a.s. < s < t. : (1 B t N(, t (2 s < t 1.4 (3 B t B s F s E[B t B s = E[(B t B s B s + E[B 2 s = E[B t B s E[B s + s = s. (3 (2 A F s E[(B t B s 1 A = E[B t B s P (A =. 1.7 B B a, B b, B c Brown s >, γ > (1 Bt a := B t+s B s, t. (2 Bt b := B t, t. (3 Bt c := γb t/γ 2, t. 8

9 : 1.2 (1.2 (1 ( = t < t 1 < < t n, g C b (R n E[g(Bt a 1,..., Bt a n = E[g(B t1 +s B s,..., B tn +s B s n = g(x 1 x,..., x n x p(s, x p(t i t i 1, x i x i 1 dx dx 1 dx n R n+1 i=1 n = p(s, x dx g(x R 1,..., x n p(t i t i 1, x i x i 1 dx 1 dx n n R = E[g(B t1,..., B tn. i=1 1.3 B = (B t t d Brown A d d AB t d Brown : B = (B t t d Brown (1.1 p(t, x p(t, x = 1 (2πt d/2 e x 2 /2t, t >, x R d (1.2 A 1,..., A n B(R d B t t [T 1, T 2, T 1 < T 2 a.s.ω : 1.7 T 1 =, T 2 = 1 [, 1 : = t < t 1 < < t N = 1 V (2 (ω = N j=1 {B t i (ω B ti 1 (ω} 2 lim E[(V (2 12 = = max 1 i N {t i t i 1 } Z i := {B ti (ω B ti 1 (ω} 2 (t i t i 1 {Z i } 1 i N E[Z i = ( E[(V (2 N 2 N 12 = E[ Z i = E[Z 2 i = j=1 j=1 N 2(t i t i j=1 1 in L2 { k } k 1 a.s. δ (ω = max i B ti (ω B ti 1 (ω B t (ω [, 1 lim δ (ω =. V (ω = sup V (ω, V (ω = N i=1 B t i (ω B ti 1 (ω { k } V (2 V (2 V (ω V k (ω V (2 k (ω δ k (ω a.s. Brown path [KS p.19, p.112, p.114 9

10 1.9 (1 a.s.ω B t (ω t (2 : lim sup t B t 2t log log 1/t = 1 a.s. lim inf t B t 2t log log 1/t = 1 a.s. lim sup t (3 Hölder : B t 2t log log t = 1 a.s. lim sup δ 1.4 Markov 1 2δ log 1/δ lim inf t B t 2t log log t = 1 a.s. max B t B s = 1 a.s. s<t 1:t s δ B = (B t t d Brown x R d Brown (x+b t t (W d, B(W d P x P x E x [ (W d, B(W d, P x B t (w, w W d B t (w = w t 1.4 (1 Y = Y (w W d R d x E x [Y R ( Y (w = n k=1 f k(w tk, t 1 < < t n f k C b (R d (2 Y = Y (w: W d B(W d - R d x E x [Y R B(R d - B(W d F t := σ(b s ; s t, F t := F t N, F t := s>t F s N := { N B(W d ; P x (N = } (F t F t x W d θ s : W d W d, s (θ s w t := w t+s B t θ s = B t+s 1.1 (F t Markov x R d, s Y = Y (w: W d B(W d - E x [Y θ s F s = E Bs(w[Y, P x -a.s. w W d (1.5 (1.5 ( s s ( Brown : (1.5 F s - Z : E x [Y θ s Z = E x [E Bs(w[Y Z ( well-defined 1

11 1 : f : R d R Borel Y = f(b t, t f(x = e iξ x, ξ R d ( f 1 Y = e iξ B t E x [e iξ B t θs Z = E x [e iξ B t+s Z = E x [e iξ (B t+s B s e iξ Bs Z = E x [e iξ (B t+s B s E x [e iξ Bs Z = e ξ 2 t/2 E x [e iξ Bs Z E y [e iξ Bt = E [e iξ (Bt+y = e iξ y e ξ 2t/2 2 : Y = n k=1 f k(b tk, t 1 < < t n f k : R d R Borel n n = 1 1 n 1 E x [Y θ s Z = E x [ n k=1 = E x [ E Bs [ [ f k (B tk +s Z = E x E Bt1 +s E Bt1 [ n k=2 = E x [ E Bs [f 1 (B t1 [ n k=2 f k (B tk t 1 f 1 (B t1 n k=2 f k (B tk t 1 f 1 (B t1 +sz Z f k (B tk Z = E x [E Bs [Y Z 1 2 f 1 (B t1 +sz F t1 +s f 1 (B t1 F t1 - ( 3 : Y 1.1 (1 B(W d = B K (W d 2 Y 1 ( (Ft Markov F t = F t, t (1.6 B(W d - Y Fs - Z : 1 : (1.6 F s - Z Y = f(b t, f C b (R d, t Z F s+ε -, ε > 1.1 E x [f(b t+s+ε Z = E x [E Bs+ε [f(b t Z. ε f B t Lebesgue E x [f(b t+s Z g t (y = f(zp(t, z y dz R d g t E x [E Bs+ε [f(b t Z = E x [g t (B s+ε Z Lebesgue E x [g t (B s Z = E x [E Bs [f(b t Z 2 : 1 Borel f : R d R Fs - Z Y = n k=1 f k(b tk, t 1 < < t n f k Borel (1.6 11

12 3 : Y 2 t i s < t i+1 2 Fs - Z E x [Y Z = E x [Y 1 θ s Y 2 Z = E x [E Bs [Y 1 Y 2 Z, Y 1 = n k=i+1 f k(b tk s, Y 2 = i k=1 f k(b tk E Bs [Y 1 Y 2 F s - Y F s - Ỹ E x [Y 1 A = E x [Ỹ 1 A, A F s (1.7 Y : B(W d - (cf. 1.5 Y = 1 G, G Fs E[(1 G 1 G 1 A =, A Fs 1 G 1 G F s 1 G 1 G =, P x -a.s. N F s 1 G F s - G F s i.e., Fs F s 1.5 Y : B(W d - F s Ỹ (1.7 : 2 {Y n } Y L 1 (P x - E[(Ỹm Ỹn 1 A E[ Y m Y n A = {Ỹm Ỹn > }, {Ỹm Ỹn < } {Ỹn} L 1 (P x Chauchy Ỹ {Ỹn} a.s. N F s Ỹ F s (Blumenthal -1 A F (= F P x (A = 1 : A F P x (A = E x [1 A = E x [1 A θ 1 A = E x [E B [1 A 1 A = E[P x (A1 A = P (A 2. P x (A = Brown σ (, := inf{ t > ; B t (, } (Brown P (σ (, = = 1 A := {σ (, = } F 1.12 P (σ (, = = or 1. t > P (σ (, t P (B t = 1/2 t P (σ (, = 1/2 ( Brown 12

13 1.5 Brown Fourier 1.2 {ξn} k L 2 [, 1 {ψn} k : ψ 1 (t 1 } ψn(t k = 2 {1 (n 1/2 [ k 1 2 n, k 2 n (t 1 [ k 2 n, k+1 2 n (t, k I(n, n N I( = {1}, I(n = {1, 3,..., 2 n 1} (n N B (N t (ω = N n= k I(n ξ k n(ω ψ k n(s ds (1.8 ( L 2 [, 1 ( {ψn; k k I(n, n =, 1,... } L 2 [, 1 : g L 2 [, 1 k I(n, n =, 1,... g(tψk n(t dt = g = a.e. n N (k+1/2 n k/2 n g(t dt =, k =, 1,..., 2 n 1 (1.9 { k/2 n ; k =, 1,..., 2 n, n N } [, 1 1 n = g(t dt = 1 g(tψ1 (t dt = n 1 n k k = k/2 2 n (t = 1 ( 1[ k 2 2 n 1, k (t + 2 (n 1/2 ψ k+1 2 n 1 n (t g(t dt = 1 ( (k +1/2 n 1 1 g(t dt 2 (n 1/2 g(tψ k k/2 2 n(t dt =. n 1 [ k 2 n, k+1 (k+1/2 n k /2 n 1 (k+2/2 n (k +1/2 n 1 (k+1/2 n g(t dt = g(t dt g(t dt =. (k+1/2 n k /2 n 1 k/2 n 1.14 a.a.ω (ω; t 1} N [, 1 {B (N t : b n (ω = max k I(n ξn(ω k x > 2 P ( ξn k > x = π ( P (b n > n = P x k I(n e u2 2 u 2 du π x x e { ξn k > n} 2 n P ( ξn 1 > x 1 u 2 2 du = 2 π 2 π e x2 2 x 2 n e x2 2 x, n 1. n P (b n > n < Borel-Cantelli Ω = N=1 n=1 {b n n} P ( Ω = 1 ω Ω N(ω n N(ω b n (ω n 13

14 max t 1 n=n(ω+1 k I(n ξ k n(ω ψ k n(s ds n=n(ω+1 n2 n n <. (1.8 (ω; t 1} [, 1 {B (N t 1.2 : 1 : 1.14 B t = lim N B (N t (B t t [,1 Brown t B t (ω 1.14 (cf. 1.3 = t < t 1 <... < t m 1 {B tj B tj 1 } m j=1, t j t j 1 : λ j R, j = 1,..., m [ { m } m { E exp i λ j (B tj B tj 1 = exp 1 } 2 λ2 j(t j t j 1. j=1 λ m+1 =, Sn(t k = ψk n(s ds {ξn} k N(, 1 [ { m } [ { m } E exp i λ j (B (N t j B (N t j 1 = E exp i (λ j+1 λ j B (N t j [ { = E exp = = N j=1 n= k I(n N n= k I(n [ = exp 1 2 i N ξn k n= k I(n j=1 [ { E exp iξ k n j=1 j=1 j=1 m } (λ j+1 λ j Sn(t k j m } (λ j+1 λ j Sn(t k j j=1 [ exp 1 { m } 2 (λ j+1 λ j S k 2 n(t j m j=1 j =1 m (λ j+1 λ j (λ j +1 λ j N n= k I(n Sn(t k j Sn(t k j Parseval n= k I(n Sk n(t j Sn(t k j = 1 1 [,t j 1 [,tj dt = t j t j N ( Lebesgue [ { m } [ E exp i λ j (B tj B tj 1 = exp 1 m m (λ j+1 λ j (λ j 2 +1 λ j t j t j 2 : { = exp { = exp j=1 m m j=1 j =j+1 j=1 j =1 (λ j+1 λ j (λ j +1 λ j t j 1 2 m (λ j+1 λ j ( λ j+1 t j 1 2 j=1 { m 1 1 = exp (λ 2 j+1 λ 2 2 jt j 1 2 λ2 mt m }= j=1 m } (λ j+1 λ j 2 t j j=1 m } (λ j+1 λ j 2 t j j=1 m j=1 { exp 1 } 2 λ2 j(t j t j 1. p

15 2 Martingales [F1 (Ω, F, P X F σ- G Q(B := E[X, B X(ωP (dω, B G B Q (Ω, G Q P Radon-Nikodym ( 1.6, 1.7 G- Y Q(B = Y (ωp (dω, B G B 2.1 (1 Y E[X G G X (2 A F P (A G := E[1 A G A E[X G P (A G 2.1 X, Y, X n, n N (1 a, b R E[aX + by G = ae[x G + be[y G a.s. (2 X a.s. E[X G a.s. (3 X G- XY E[XY G = XE[Y G a.s. E[X G = X a.s. (4 H, G F σ- H G E[E[X G H = E[X H a.s. E[E[X G = E[X (5 (Jensen ψ R ψ(x ψ(e[x G E[ψ(X G a.s. (6 X n X in L 1 E[X n G E[X G in L 1 (7 X G E[X G = E[X a.s. A 1, A 2 F P (A 1 A 2 G = P (A 1 G + P (A 2 G a.s. A 1, A p = { p(ω, A ; ω Ω, A F } p G P (1 ω Ω p(ω, (Ω, F 15

16 (2 A F p(, A G- (3 P (A B = p(ω, A P (dω, A F, B G. B (Ω, F 7 Ω F = B(Ω [IW p.13, [SV p.12 [SV conditional probability distribution regular 2.2 (Ω, F P (Ω, F G F σ- G P p = { p(ω, A ; ω Ω, A F } H G σ- G- N ω / N p(ω, A = 1 A (ω, A H 2.2 Stopping time (F t t (Ω, F, P filtration F (F t t 2.3 σ : Ω [, (F t t stopping time ( Markov {σ t} { ω ; σ(ω t } F t, t 2.3 (i (iv (i σ stopping time. (ii {σ < t} F t, t. (iii {σ > t} F t, t. (iv {σ t} F t, t. : (i (iii, (ii (iv (i (ii (ii (i {σ < t} = n=1{σ t 1/n} F t {σ t} = n=n{σ < t + 1/n} F t+1/n N N (F t 7 (Ω, F {1,..., n}, N, [, 1 Borel Ω (σ- Borel ϕ ϕ, ϕ 1 16

17 2.4 σ, τ, σ n, n N stopping time (i σ + τ, σ τ, σ τ stopping time. (ii σ n σ n+1, n N σ n σ n+1, n N lim n σ n stopping time. : Q t = (Q [, t {t} (i {σ + τ t} = n=n r Q t [{σ r} {τ t r + 1/n}, N N. {σ τ t} = {σ t} {τ t}, {σ τ t} = {σ t} {τ t}. (ii {σ n } { lim σ n t} = n n=1{σ n t}, {σ n } { lim σ n < t} = n n=1{σ n < t}. 2.1 (X t (Ω, F, P (F t - d X E R d (first hitting time σ E (ω = inf{ t > ; X t (ω E } (2.1 E σ E stopping time (Ω, F, P E B(R d σ E stopping time 8 : E E {σ E < t} = r Q (,t {X r E} F t E E n = { x R d ; d(x, E < 1 } n E n σ En stopping time. E n E σ En (ω σ E (ω, ω Ω 2.4 (ii σ E stopping time 2.4 stopping time σ F σ = { A F ; t A {σ t} F t } ( σ stopping time (2.2 F σ σ- σ t ( (2.2 F σ F t 2.5 σ, τ, σ n, n N stopping time (i σ F σ - (ii σ τ F σ F τ. (iii ( ω Ω σ n (ω σ(ω n=1f σn = F σ. : (i: {σ s} F σ, s {σ s} {σ t} F t, t {σ s} {σ t} = {σ t s} F t s F t (ii: A F σ A {σ t} F t, {σ t} {τ t} A {τ t} = [A {σ t} {τ t} F t, t A F τ 8 p.51 C.Dellacherie: Capacités et Processus Stochatiques. Springer,

18 (iii: (ii A n=1f σn A {σ < t} = n=1a {σ n < t} F t, t A F σ (cf. 2.3 (ii (i 2.3 Martingales (Ω, F, P F filtration (F t t 2.5 ( X = (X t t (F t t (martingale (i t E[ X t <. (ii X = (X t t (F t -adapted. (iii s t E[X t F s = X s a.s. 3 (iii E[X t F s X s a.s. X (submartingale, E[X t F s X s a.s. X (supermartingale Brown B = (B t Ft B = σ{ B s ; s t } N 1.11 (Ft B F 1.6 (3 (1 (2, (3 (1 (B t (F t B (2 X t = Bt 2 t (X t (F t B (3 σ R Y t = e σb t σ2 2 t (Y t (F t B 2.3 p 1 X = (X t p ( X t p 2.6 (Doob X = (X t λ >, p > 1 ( (i P sup X s λ 1 [ s t λ E X t, sup X s λ. s t (ii t X t (ω a.s. ( P sup s t sup s t X s λ 1 [X λ E t (iii X E[ X t p <, t [ E X s p p p 1 E[ X t p : t k = kt/n (X tk k [F1, p ( P sup 1 k n X tk λ 1 [ λ E X t, sup X tk λ 1 k n (X t n (i (ii (i (iii Y = sup s t X s 18

19 E[Y p = p = p = λ p 1 P (Y λ dλ p λ p 2 dλ Ω p p 1 E[Y p 1 X t λ p 1 1 λ E[ X t, Y λ dλ ((i X t 1 {Y λ} dp (Fubini p p 1 E[Y (p 1/p E[ X t p 1/p (Hölder E[Y (p 1/p p (iii 2.7 (Doob (optional sampling theorem σ, τ stopping time σ τ X = (X t E[X τ F σ X σ a.s. (2.3 X = (X t (2.3 X τ X τ (ω = X τ(ω (ω X σ : σ n (ω = [nσ(ω+1, τ n n (ω = [nτ(ω+1 σ, τ σ n n, τ n [F1, p E[X τn F σn X σn a.s. F σ F σn A F σ E[X τn, A E[X σn, A σ n (ω σ(ω, τ n (ω τ(ω (X t {X τn } n, {X σn } n [F1, p lim n E[X τ n, A = E[X τ, A, lim n E[X σn, A = E[X σ, A X E[X τn E[X τn+1, n N inf n E[X τn E[X > ε > N N E[X τn < inf n E[X τn + ε λ > n N E[ X τn, X τn > λ = E[X τn, X τn > λ + E[X τn, X τn λ E[X τn E[X τn, X τn > λ + E[X τn, X τn λ E[X τn + ε E[ X τn, X τn > λ + ε, P ( X τn > λ 1 λ E[ X τ n = 1 λ (2E[X τ n E[X τn 1 λ (2E[X inf n E[X τ n sup n N P ( X τn > λ (λ {X τn } n {X σn } n σ, τ 1 (X t σ :=, τ := σ {1} ( 1 τ < a.s. ( Brown cf. 2.2 X σ =, X τ = 1 a.s

20 R a, b 2.8 M = (M t M = x (a, b τ := σ [a,b c < a.s. P (M τ = a = b x b a, P (M τ = b = x a b a. (2.4 : M x = E[M τ t, t > M τ t a b Lebesgue t τ < a.s. x = E[M τ = ap (M τ = a + bp (M τ = b τ < a.s. P (M τ = a + P (M τ = b = 1. ( x (a, b 1 Brown B = (B t τ := σ [a,b c < a.s. c R σ {c} < a.s. : 2.2 (ii Bt 2 t E[τ t = E[B 2 τ t ( a b 2, t t Fatou E[τ ( a b 2 τ < a.s. (2.4 b P (B σ{a} = a = 1, a P (B σ{b} =b = 1 c R σ {c} < a.s. 2.4 Brown Markov d Brown 1.4 x R d Brown (W d, B(W d P x ( B t (w = w t, w W d, t filtration (F t (F t N = {N B(W d ; P x (N =, x R d } (F t stopping time σ σ θ σ : W d {σ < } W d (θ σ w t := w t+σ(w, t 1.1, ( Markov x R d Y = Y (w: W d B(W d - E x [Y θ σ F σ = E Bσ [Y, P x -a.s. w {σ < }. (2.5 : 1.1, 1.11 f C b (R d Y = f(b t (2.5 E x [f(b t+σ, A {σ < } = E x [E Bσ [f(b t, A {σ < }, A F σ ( N σ σ n = ([nσ + 1/n σ n stopping time A F σ 2

21 Ex [f(b t+σn, A {σ < } = = E x [f(b t+σn, A {σ n = k/n} (2.6 k=1 E x [E [f(b Bσ n t, A {σ n = k/n} = E x [E [f(b Bσ n t, A {σ < }. (2.7 k=1 2 A F σ F σn A {σ n = k/n} F k/n Brown Markov {σ < } σ n σ f, B t Lebesgue (2.6 E x [f(b t+σ, A {σ < } (1.1 E [f(b Bσ n t = f(xp(t, x B σn dx R d f(xp(t, x B σ dx = E Bσ [f(b t R d (2.7 E x [E Bσ [f(b t, A {σ < } 2.3 ( 1 Brown B t m t := sup s t B s a, x, t > P (B t < a x, m t a = P (B t > a + x. : a σ := σ {a} stopping time Markov P (B t > a + x = P (σ t, B t > a + x = E[P Bσ (B t σ > a + x, σ t = E[P Bσ (B t σ < a x, σ t = P (σ t, B t < a x = P (B t < a x, m t a. 2 {σ t} B σ = a : P a (B t σ > a + x = P a (B t σ < a x Brown B t P ( sup B s a = 2P (B t a, a, t >. s t : P (m t a = P (B t < a m t + P (B t a 2.3 x = 1 P (B t > a P (B t = a = Brown B t a σ {a} := 1 a2 inf{ t > ; B t = a } e 2t, t > 2πt (Ω, F, P F filtration (F t t 2.6 X = (X t (DL a > { X σ a ; σ stopping time } 21

22 2.1 (Doob-Meyer (DL (M t A = (F t - (A t X t = M t +A t, t (A t (A t (A t t s A t A s a.s. (M t E[ M (,t sda s = E[ M (,t s da s (M t X t = Mt 2 (X t Doob- Meyer 9 M t (X t stopping time σ [ E[ X σ a, X σ a λ E[ X a, X σ a λ E X a, sup X t λ. t a Doob (ii P (sup t a X t λ 1 E[ X λ a (X t (DL A t : (A t A = a.s., (F t -adapted A t1 A t2 < a.s., t 1 < t 2 A +,c A c := { A 1 A 2 ; A 1, A 2 A + } M p c := { M = (M t ; M E[ M t p <, t } p 1, A A c a.a. ω Ω A t (ω ( 2.7 M M 2 M M A +,c M M 2 (quadratic variation 2.11 M = (M t M M 2 c A c M t = M, t a.s. : M = : = t < t 1 < < t n = t = max 1 i n {t i t i 1 } E[M 2 t = n E[Mt 2 i Mt 2 i 1 = i=1 E [ max i M ti M ti 1 n E[(M ti M ti 1 2 i=1 n M ti M ti 1 i=1 M t = A 1 t A 2 t A c, A 1, A 2 A +,c σ k := inf{ t > ; A 1 t + A 2 t > k } (inf = 9 [N : [KS p

23 2.1 σ k stopping time 2.7 Mt k := M t τk n i=1 M t k i Mt k n i 1 i=1 ( A1 t i τ k A 1 t i 1 τ k + A 2 t i τ k A 2 t i 1 τ k k ω Ω Ms k (ω [, t E[(Mt k 2 ke[max Mt k i i Mt k i 1 as. Mt k = a.s. k τ k (k M t = a.s. : = t < t 1 < < t n < t n+1 < t n < t t n+1 Q t (M; := n (M ti M ti (M t M tn 2, Q (M; = i=1 Brown 1.8 Q t (B; t in L (2 Bt 2 t 2.12 M M 4 c M = ( M t A +,c Q t (M; M t in L 2, t M 2 M M 2 A A A +,c 2.13 M M 4 c C > E[Q t(m; 2 C : M t K, K > : = s < s 1 < < s l < t < s l+1 = t { l } 2 Q t (M; 2 = (M sk+1 M sk 2 k= l = sk+1 M sk k=(m (M sj+1 M sj 2 (M sk+1 M sk 2 j<k l = sk+1 M sk k=(m (M sj+1 M sj 2 (Q t (M; Q sj+1 (M; j l = sk+1 M sk k=(m (Q sj+1 (M; Q sj (M; (Q t (M; Q sj+1 (M; j E[Q t (M; Q sj+1 (M; F sj+1 = E[(M t M sj+1 2 F sj+1 E[Q t (M; 2 [ l = E (M sk+1 M sk j k= E[ (max E[(Q sj+1 (M; Q sj (M; (M t M sj+1 2 j l M s j+1 M sj max j l M t M sj+1 2 Q t (M; 12K 2 E[Q t (M; = 12K 2 E[M 2 t M 2 12K 4. 23

24 Mt = sup s t M s max j M sj+1 M sj 2 4(Mt 2 E[Q t (M; 2 E[12(M t 2 Q t (M; E[12 2 (M t 4 1/2 E[Q t (M; 2 1/2. 2 E[Q t (M; 2 Doob (iii E[Q t (M; E[(M t (4/3 4 E[M 4 t < t := M τk t, τ K = inf{ s ; M s > K } Fatou M (K 2.14 Y (n = (Y s (n s [,T (F t -adapted [ lim E sup Y s (n Y s (m 2 = n,m k s T s T (F t -adapted (Y s {Y (n } n {Y (nk } k ( [ P lim sup Y (n k s Y s = = 1 lim E Y (n k s Y s 2 = k sup s T [ : n k n k+1 E sup s T Y s (n Y s (m 2 < 1/2 3k, n, m > n k { A k := sup s T Y (n k+1 s Y (n k s > 1 2 k } Chebyshev [ P (A k 2 2k E P (Ω = 1. sup s T Y (n k+1 s Y (n k s 2 < 1 2 k. k P (A k < Borel-Cantelli Ω = m=1 k=m Ac k ω Ω m = m(ω k m n j n i m sup s T Y (n k+1 s Y (n k s 1 2 k sup Y (n i s Y (n j s s T j 1 l=i l 2. i 1 Y (nk C([, T Cauchy Y = (Y s sup s T Y (n k s Y s, k. [ [ lim E sup Y (n k s Y s 2 = lim E lim sup Y (n k s Y (n j s 2 k s T k j s T [ lim lim E Y (n k s Y (n j s 2 = k j sup s T Fatou 24

25 2.12 : ( Mt 2 A t, Mt 2 Ãt A, Ã A +,c A t Ãt = (Mt 2 Ãt (Mt 2 A t M 2 c A c 2.11 A t Ãt = A Ã =, t a.s. ( Q t (M; L 2 - Cauchy : = t < t 1 < < t n < t n+1 <, t n < t t n+1 t k < s t k+1 E[(M tk+1 M tk 2 F s = E[(M tk+1 M s 2 F s + (M s M tk 2 E[Q t (M; F s = = M 2 t k (M ti+1 M ti 2 + (M s M tk 2 i= +E[(M tk+1 M ts 2 + n 1 i=k+1 (M ti+1 M ti 2 + (M t M tn 2 F s k (M ti+1 M ti 2 + (M s M tk 2 + E[Mt 2 Ms 2 F s. i= E[Q t (M; Q s (M; F s = E[M 2 t M 2 s F s (2.8 Q t (M; Mt 2 Q t (M; L t := Q t (M; Q t (M; 2 (2.8 M L,, s = E[Q t (Q(M; Q(M; ; = E[L 2 t L 2 (2.9 Q t (Q(M; Q(M; ; 2Q t (Q(M; ; + 2Q t (Q(M; ; : s < s 1 < < s l < s l+1, s l t < s l+1 t j(k s k < s k+1 t j(k+1 Q sk+1 (M; Q sk (M; = (M sk+1 M tj(k 2 (M sk M tj(k 2 = (M sk+1 M sk (M sk+1 + M sk 2M tj(k Q t (Q(M; ; = k (Q sk+1 (M; Q sk (M; 2 + (Q t (M; Q sl (M; 2 = k (M sk+1 M sk 2 (M sk+1 + M sk 2M tj(k 2 + (M t M sl 2 (M t + M sl 2M tn 2. Q t (Q(M; ; sup M sk+1 + M sk 2M tj(k 2 Q t (M; k (2.9 25

26 E[(Q t (M; Q t (M; 2 2E[Q t (Q(M; ; + 2E[Q t (Q(M; ; 2E[sup M sk+1 + M sk 2M tj(k E[Qt (M; k +2E[sup M sk+1 + M sk 2M t j(k E[Qt (M; k {t j} sup k M sk+1 + M sk 2M tj(k (Mt 4 Doob (iii Mt := sup s t M s 4 Lebesgue lim E[sup M sk+1 + M sk 2M t, j(k 4 = 2.13 k lim E[(Q t(m; Q t (M; 2 =, Cauchy t > n n, n (Q s (M; n Q s (M; m s [,t n, m E[ sup Q s (M; n Q s (M; m 2 4E[ Q s (M; n Q s (M; m 2 s t Q s (M; n { nk } 2.14 (F t -adapted M ( [ P lim sup Q s (M; nk M s = = 1, lim E Q s (M; nk M s 2 = k k s T sup s T 2 ( M s s T [, T s Q s (M; n s M s M A +,c Ms 2 Q s (M; n Ms 2 M s 2.5 M σ stopping time Mt σ = M σ t M σ M M 4 c M σ t = M σ t 2.7 (F t -adapted X = (X t τ n τ n+1, n, lim n τ n = ( t T lim n τ n = T stopping times {τ n } n (X τ n t X = (X t M loc, M c,loc 2.15 M M c,loc (1 M A +,c M 2 M M 26

27 (2 t > [, t { n } ( lim P sup Q s (M; n M s > ε =, ε >. (2.1 n s t (3 M = M M 2 c E[ M t <, t E[M t 2 = E[ M t : (1 stopping time {τ n } M τn σ n := inf{t; M t n} σ n M n := M ρ n ρ = σ n τ n 2.12 A n A +,c (M n 2 A n ((M n+1 2 A n+1 ρn = (M n 2 (A n+1 ρn (A n+1 ρ n = A n. M t := lim n A n t, t > : (M ρn t 2 M ρ n t = Mρ 2 n t M ρn t ρn = (Mt 2 A n t (1 (2 ρ n δ > P (ρ m t < δ m M ρm Q s (M; = Q s (M ρ m ;, s ρ m ( P Q s (M; M s > ε sup s t ( lim sup P M s = M ρm s, s ρ m. ( P (ρ m t + P sup s t sup s t Q s (M; M s > ε δ δ (2 (3 M M 2 c, M = E[(M ρ n 2 M ρ n t = t E[ M t = lim n E[ M ρ n t = lim n E[(M ρn t 2 E[M 2 t. Q s (M ρ m ; M ρ m s > ε (Mt 2 Fatou E[M 2 t lim inf n E[(M ρ n t 2 = lim inf E[ M ρ n n t = E[ M t 2.2 M M 2 c M A +,c M t 2 M t 2.16 M, N M c,loc M, N A c (1 MN M, N 27

28 (2 t > [, t { n } ( lim P sup Q s (M, N; n M, N s > ε =, ε >. n s t Q s (M, N; n = i:t i+1 <s (M t i+1 M ti (N ti+1 N ti +(M s M tn (N s N tn, : = t < t 1 < < t n <, t n s < t n+1 : M, N = 1 ( M + N M N M, N, L M c,loc (i M, N = N, M. (ii M + N, L = M, L + N, L. (iii a am, N = a M, N. (iv τ stopping time M τ, N τ t = M τ, N t = M, N τ t. : 2.15, 2.16 T > M T := { M = (M t t [,T ; M (F t - E[MT 2 < } 2.18 M T E[, T Hilbert : (, := E[, T , 2.16, 2.17 ( {M n } n M T E[ M n M m T (m, n Doob (iii 2.2 [ E Ms n Ms m 2 4E[ MT n MT m 2 = 4E[ M n M m T sup s T 2.14 (F t -adapted M E[sup s T Ms n M s 2, n ( M M n 2.1 (6 E[ M n M T = E[ M n T M T 2 E[, T M T Hilbert 1 M, N, L M T, α R : (M, M, (M, M = M = a.s. : (M + N, L = (M, L + (N, L, (αm, N = α(m, N, : (M, N = (N, M (cf

29 3 Brown 1.8 f s db s Stieltjes (F s - (M s f s s (F t - (cf. 1.2 f s dm s (Ω, F, P F filtration (F t t : p 1, T > M p c,t := { M = (M t t [,T E[ M T p < }, M p c := { M = (M t t ; M E[ M t p <, t } p = 2 M T = M 2 c,t, M = M2 c filtration(f t M T (F t, M(F t 2.18 (2 3.1 (1 M T M T := E[MT 21/2 Hilbert (2 M d(m, N := k= ( M N k 1/2 k : (1 M T = E[ M T 1/ (2 {M n } M d(m n, M m (m, n T N {(Mt n t [,T } M T Cauchy (1 M (T M T l T M (T t = M (l t ( t T a.s. M t = M [t+1 t M M d(m n, M (n 2.15 M M M A +,c M 2 M M M L 2 T ( M := { f ; f = f t (ω E[ fs 2 d M s < } T T L 2 ( M := { f ; f = f t (ω E[ fs 2 d M s <, T > } ( 1/2, M M f LT := E[ fs 2 d M s L 2 T = L2 T ( M, L2 = L 2 ( M 3.2 M M (1 L 2 T ( M L T Hilbert (2 L 2 ( M ρ(f 1, f 2 := k=1 ( f 1 f 2 L 2 k 1/2 k 29 T

30 : (1 m(a = E[ T 1 A(s, ω d M s, A B([, T F T ( σ- L 2 ([, T Ω, B([, T F T, dm 3.2 L 2 T ( M ( (2 3.1 (2 f t (ω = k j=1 ξ j 1(ω1 (tj 1,t j (t; = t < t 1 < < t k, k N ξ j F tj - f L 3.3 L L 2 ( M : f L 2 ( M Lebesgue f s f (s u(ω d M 1 n u M s M (s 1 n f s (ω = lim sup fs n (ω, fs n (ω := n f s fs n F s- : = t < t 1 < < t k k fs n, (ω = ft n j 1 (ω1 (tj 1,t j (s j=1 f n, L f n lim fs n, (ω = fs n (ω, (s, ω. T E[ fs n, fs n 2 d M s,. d M a.s. s lim n fs n (ω f s (ω T f s (ω d M s = T f s (ω d M s (cf. 2.4 E[ T f n s f s 2 d M s f n, f 3.1 f s (ω = k j=1 ξ j 1(ω1 (tj 1,t j (s L I(f t := ξ j 1 (M tj M tj 1 + ξ l (M t M tl, t l < t t l+1 (3.1 j:t j <t I(f t = f s dm s (3.1 ξ j F tj I(f M (1 I(f t = f 2 s d M s. (2 E[I(f 2 t = E[ f 2 s d M s. : I(f : f t = ξ 1 (ω1 (t1,t 2 (t, ξ 1 F t1 -, s t 1 E[I(f t F s = ξ 1 (E[M t t2 F s M t1 = ξ 1 (M s M t1 = I(f s. s < t 1 E[I(f t F s = E[ξ 1 (E[M t t2 F t1 M t1 F s =, I(f s =. 3

31 2 (ξ j (1: s t j < t k E[ξ j (M tj+1 M tj ξ k (M tk+1 M tk F s = E[ξ j (M tj+1 M tj ξ k E[M tk+1 M tk F tk F s = t m s < t m+1 t l t < t l+1 E[(I(f t I(f s 2 F s { l 2 Fs = E[ ξ j 1 (M tj M tj 1 + ξ l (M t M tl + ξ m (M tm+1 M s } j=m+2 = E[ξ 2 m(m tm+1 M s 2 F s + = E[ξ 2 m(m 2 t m+1 M 2 s F s + l j=m+2 l j=m+2 = E[ξ 2 m( M tm+1 M s F s + = E[ s f 2 u M u. E[ξ 2 j 1(M tj M tj 1 2 F s + E[ξ 2 l (M t M tl 2 F s E[ξ 2 j 1(M 2 t j M 2 t j 1 F s + E[ξ 2 l (M 2 t M 2 t l F s l j=m+2 E[ξ 2 j 1( M tj M tj 1 F s + E[ξ 2 l ( M t M tl F s I(f E[{I(f t I(f s } 2 F s = E[I(f 2 t I(f 2 s F s I(f 2 t fu M 2 u (1 (2 (1 s = 3.2 f L 2 ( M f n L T lim E[ fs n f s 2 d M s =, T > n 3.4 (2 {I(f n } n M Cauchy 3.1 X M d(i(f n, X (n X X t = I(f t = f s dm s f M (stochastic integral I(f f {f n } 3.2 f L 2 ( M E[I(f 2 T = E[ I(f T = E[ : L T f 2 s d M s 3.5 M, N M, f L 2 ( M, g L 2 ( N, t > ( 1/2 ( 1/2, f s g s d M, N s fs 2 d M s gs 2 d N s a.s. (3.2 : 2.17 M + rn = M + 2r M, N + r 2 N, r Q s2 s 1 d M + rn s = s2 s 1 d M s + 2r s2 s 1 d M, N s + r 2 s2 s 1 d N s, a.s. 31

32 r r R s2 ( s 2 1/2 ( s 2 1/2. d M, N s d M s d N s s 1 f = i ξ i 11 (ti 1,t i, g = i η i 11 (ti 1,t i L i f s g s d M, N s = ξ i 1 η i 1 d M, N s i t i 1 ( i 1/2 ( i 1/2 ξ i 1 η i 1 d M s d N s i t i 1 t i 1 ( i 1/2 ( i ξ i 1 2 d M s η i 1 2 d N s i t i 1 i t i 1 ( 1/2 ( 1/2 = fs 2 d M s gsd M 2 s a.s. s 1 f, g 3.3 L f, g Lebesgue 3.6 (1 M, N M, f L 2 ( M I(f t = I(f, N t = N M X, N t = M X I(f t = (2 I(f T = T s 1 1/2 f s dm s f s d M, N s, t > (3.3 f s dm s f s d M, N s (t > X = f s dm s I : L 2 T ( M M T T > : (1 f L 2 ( M {f n } n L f f n LT (n, T > E[ I(f n I(f T = E[ T f n s f s 2 d M s, n. E[ I(f n, N T I(f, N T = E[ I(f n I(f, N T E[ I(f n I(f 1/2 T N 1/2 T E[ I(f n I(f T 1/2 E[ N T 1/2 T ( T 3.5 (f s fs n 1/2 N d M, N s f s fs n 2 1/2 d M s T E[ T f s d M, N s T 3.4(1 I(f n, N t = f n s d M, N s, n. 32 fs n d M, N s (3.3

33 X M X, N t = f s d M, N s N M X X, N = N = X X X X =. X = X a.s. ( (1 M M, f, g L 2 ( M, a, b R (af s + bg s dm s = a f s dm s + b (2 M, N M, f L 2 ( M L 2 ( N, a, b R f s d(am + bn s = a f s dm s + b : (1 N M = a (af s + bg s dm, N t = f s dm s, N t + b (af s + bg s d M, N s = a g s dm s, N t = a g s dm s, t >. f s dn s, t >. f s dm s + b f s d M, N s + b g s dm s, N t. g s d M, N s 3.6 (2 3.5 L 2 ( M L 2 ( N L 2 ( am + bn L M = a f s d(am + bn s, L t = f s d M, L s + b f s d am + bn, L s f s d N, L s = a f s dm s + b f s dn s, L t. 3.3 ( M M c,loc (cf. 2.7 M ( T P fs 2 d M s < = 1, T > (3.4 f = (f s f s dm s 2.7 stopping time {σ n } σ n M t σn τ n = n inf{ t ; f s d M s n } τ n ρ n = σ n τ n = M t ρn, ft n (ω = f t (ω1 {ρn t}(ω M n t M n M, f n L 2 ( M n I(f n t = fs n dms n (iv I(f n t = I(f m t, t ρ n, n m ρ n (n I(f t = I(f n t, t ρ n I(f f M 33

34 3.3 ( 3.6 M M c,loc (3.4 f = (f s I(f t = f s dm s M c,loc ( M M c,loc, (3.4 (F t - f = (f s : = s < s 1 <... < s n = t ( t lim P f s dm s f(s i (M i+1 M si > δ =, δ > i 3.2 ( 11 (stochastic calculus 3.4 X = (X t X t = X + M t + A t, M M c,loc, A A c X F - M = a.s. A = a.s., X = (X t (semi-martingale 12 d 3.8 X t = (Xt 1,..., Xt d : X i t = X i + M i t + A i t, i = 1,, d. (3.5 f C 2 (R d f(x t d f d f(x t f(x = x (X s dm i f i s + x (X s da i i s i= d i,j=1 i=1 2 f x i x j (X s d M i, M j s. (3.6 : f i := f, f x i ij = 2 f x i x j τ n := inf{ t > ; X M t A t > n } inf = τ n (n X t τn (3.6 X, M t, A t f, f i, f ij : f + i f t + ij f ij K. [, t : = t < t 1 < < t n = t Taylor ξ k = Xt i k 1 + θ k (Xt i k Xt i k 1, θ k (, 1 d 11 x = (x t f C 1 (R dt f(x t = f (x t ẋ t f(x t f(x = f (x s ẋ s ds = f (x s dx s

35 f(x t f(x = = n k=1 n (f(x tk f(x tk 1 k=1 d f i(x tk 1 (Xt i k Xt i k i=1 n d k=1 i,j=1 f ij(ξ k (X i t k X i t k 1 (X j t k X j t k 1. d i=1 f x i (X s dm i s + d i=1 f x i (X s da i s ( n d k=1 i,j= n k=1 i,j=1 f ij(ξ k (M i t k M i t k 1 (M j t k M j t k 1 + d n d k=1 i,j=1 f ij(ξ k (A i t k A i t k 1 (A j t k A j t k 1 I 1 + I 2 + I 3 f ij(ξ k (M i t k M i t k 1 (A j t k A j t k 1 A i = A i,+ A i,, A i,+, A i, A +,c I 2 K sup M tk M tk 1 ( A + t + A t,, a.s. k I 3 K sup A tk A tk 1 ( A + t + A t,, a.s. k M t = (Mt 1,..., Mt d, A + t = (A 1,+ t,..., A d,+ t, A t = (A 1, t,..., A d, t I 1 (3.6 I 1 := 1 2 n d k=1 i,j=1 f ij(x tk 1 (M i t k M i t k 1 (M j t k M j t k Q t (M, I 1 I sup f ij(ξ l f ij(x tl 1 Mt i k Mt i k 1 M j t k M j t k 1 l k i,j 1 2 sup f ij(ξ l f ij(x tl 1 Q t (M m ; 1/2 Q t (M n ; 1/2 i,j,l m,n Schwarz 2 E[ I 1 I m,n m,n E [ sup f ij(ξ l f ij(x tl 1 Q t (M m ; 1/2 Q t (M n ; 1/2 i,j,l E [ sup f ij(ξ l f ij(x tl 1 2 1/2 [ E Qt (M m ; Q t (M n ; 1/2 i,j,l 1 2 E[ sup f ij(ξ l f ij(x tl 1 1/2 E [ Q t (M m ; 2 1/4 [ E Qt (M n ; 2 1/4 i,j,l m,n 35

36 2.13 I 1 := 1 2 n d k=1 i,j=1 f ij(x tk 1 ( M i, M j tk M i, M j tk 1 E[(Ms i 2 Ms i 1 (Ms j 2 Ms j 1 ( M i, M j s2 M i, M j s1 F s1 = E[ I 1 I 1 2 = 1 4 k E [( k i,j K 2 E [ M tk M tk i,j f ij(x tl 1 {(M i t k M i t k 1 (M j t k M j t k 1 ( M i, M j tk M i, M j tk 1 2 K 2 E [ sup M tk M tk 1 2 Q t (M i ; k i +K 2 E [ sup M i, M j tk M i, M j tk 1 ( M i, M j + t + M i, M j t k i,j E[ I 1 I 1 2 (. I I i,j i,j f ij(x s d M i, M j s in L1, as. k 2 d M i, M j s } t k 1 f ij(x s d M i, M j s (3.6 t a.s. t 1 t ( Brown B = (B t (F t -Brown (i B = (B t (F t - (ii s t B t B s F s 1.4 Brown F,B -Brown 3.5 B p t = p B p 1 p(p 1 s db s + Bs 2p 2 ds 2 E[B 2p t, p N ( : Brown (B t B t = t 3.1 B t x N Brown f(x = x m = ((x (x N 2 1/2, m 2 B t m x m = m N i=1 B i s B s m 2 db i s s (Nm + m(m 2 B s m 2 ds N 3 m = 2 N, σ n := { t ; B t 1/n } B t σn m x m = m N i=1 σn B i s B s m 2 db i s x σ n (n B t m N = 3, m = 1 3 Bessel 36

37 3.2 B t x N Brown f(x = x m = ((x (x N 2 1/2, m 2 B t m x m = m N i=1 B i s B s m 2 db i s (Nm + m(m 2 B s m 2 ds N 3 m = 2 N, σ n := { t ; B t 1/n } B t σn m x m = m N i=1 σn B i s B s m 2 db i s x σ n (n B t m N = 3, m = 1 3 Bessel 37