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2 , Brown,, dn(t = a(tn(t, N( = N (1 dt N(t t, a(t t, Malthus {N(t} t, (1 a(t,, a(t = r(t + ( r(t,,,, Brown,,,,, Brown, Itô Calculus,,,,,, Kalman-Bucy,, (1, s t N(s Z(s, N(s, Z(s = N(s + (2 i

3 {Z(s} s t {Z(s} s t (1 N(t?, ( 1961 Kalman Bucy Kalman-Bucy,,,,,,, 1, 2, 3,, 3,,,,,, Kalman-Bucy, 5 1, 2,8,9,1,12,13, 1 1, 2 Lebesgue, Fubini, Fourier Gauss ii

4 3 Borel-Cantelli, Lebesgue 4 Radon-Nikodym,, Hilbert 2 Brown, Brown, EX n+1 F n = X n X n+1 F n, X n 12, 1 Brown,, σ- ( 2, Brown, Doob 3 2 M, M 2 M M, M M, 3 2 Hilbert, 1, 2 M, Riemann, f(s, ω M M { f(s, ω dm s} t M f(s, ω M iii

5 2, Lebesgue-Stieltjes,,, 3, (, Brown Lévy 4,, 1, 2, Brown Ornstein-Uhlenbeck, Black- Scholes Brown 3, 5 1, Kalman Bucy 1, t X t, {Z s } s t X t X t, Hilbert 2 1 X t {Z t } 2 {N t } ( Z-, N- {N t } Brown {R t }, R t X t 3 X t Z t, Xt Kalman-Bucy, iv

6 ,,,,,,, v

7 Brown Brown vi

8 1 Ω F P Ω,, 11 Ω Ω F, i Ω F ii A F A c F iii A n F, n N (:= {1, 2, } n=1 A n F, σ-, A c A {F t } t T, T Ω σ-, t T F t σ-, Ω C C σ- {F t } t T, C σ- σ(c, σ(c = t T F t, σ- 111 Ω D, Dynkin i Ω D ii A, B D, B A A \ B D iii A n D, A n A n+1, n N n=1 A n D 112 ( Dynkin C Ω, A, B C A B C D C Dynkin σ(c D 1, p447 Ω, σ- F F-, (Ω, F, Ξ, Ξ σ- Borel, B(Ξ 1

9 1 2 (Ω, F µ i A F, µ(a ii A n F, n N, A i A j = (i j µ ( n=1 A n = n=1 µ (A n, µ (Ω, F, (Ω, F, µ µ : iii µ(ω = 1 µ P (Ω, F (Ω, F, P, P (A A (Ξ, B(Ξ, ( A 1,, A n k, A i1,, A ik P j=1 A i j = k j=1 P ( A ij {A λ }, λ Λ, {A λ }, λ Λ F σ- F 1,, F n, F i A i A 1,, A n F σ- {F λ }, λ Λ, F λ 12 (Ω, F, Ξ f : (Ω, F (Ξ, B(Ξ F-, A B(Ξ, f 1 (A := {ω Ω ; f(ω A} F, f : (Ω, F (Ξ, B(Ξ, f Ω Ξ F B(Ξ, f, r f 1( (, r F, R N ( N Euclid B(R N - N Borel 121 ( (Ω, F A F, { 1 (ω A I A (ω := (ω A c, I A F- I A 122 Ω i f(ω, g(ω, α, β αf(ω + βg(ω f(ωg(ω

10 1 3 ii f n (ω, n = 1, 2, sup f n (ω, inf n 1 n 1 f n(ω, lim sup n f n (ω, lim inf n f n(ω f(ω = lim f n (ω f(ω n 8, pp (X, B(X, X R N Z = X R N σ- B(Z = B(X B(R N Z f(x, y, x X, y R N i y x B(X-, ii x y f(x, y (x, y Z B(Z- 8, pp (Ω, F, P X : (Ω, F (Ξ, B(Ξ F-, X Ξ- Ξ Euclid, X = (X 1,, X N N N F {X 1 (A ; A B(R N } σ- X σ-, σ(x σ(x X σ- X 1,, X n, σ(x i, i = 1,, n, A i (X i, P (X 1 A 1,, X n A n = n P (X i A i i=1 {X λ }, λ Λ, σ(x λ, λ Λ, A λi (X λi i = 1,, n, P (X λ1 A λ1,, X λn A λn = n P (X λi A λi i=1

11 1 4 {X λ }, λ Λ, λ Λ σ(x λ σ- σ(x λ ; λ Λ {X λ }, λ Λ {Y γ }, γ Γ, σ(x λ ; λ Λ σ(y γ ; γ Γ X, Y, φ, ψ Borel, X Y φ(x ψ(y, (Ω, F, µ f(ω, µ f(ωµ(dω f(ω dµ(ω Ω f(ω f(ω µ(dω < Ω, f, f 125 X (Ω, F, P X P, EX := X(ωP (dω X V (X := E (X EX 2 A F, EX ; A := EXI A (ω = X(ωP (dω Ω N X = (X 1,, X N, N M := (EX 1,, EX N, Cov(X i, X j := E(X i EX i (X j EX j X i X j, N N V := (Cov(X i, X j X (Ω, F, P X, X L 1 (Ω, F, P ( L 1 (Ω, L 1 (P L 1 X L p (Ω, F, P ( L p (Ω, L p (P, L p, E X p = X(ω p P (dω < Ω L 2 (Ω, F, P 126 X L 2 (Ω, F, P ( X := X(ω 2 P (dω Ω A Ω 1 2 = E X(ω 2 1 2, L 2 (Ω, F, P Hilbert

12 1 5 8, p163 13, pp91 93,p X, Y (Ω, F, P 1 i ( Hölder p > 1, + 1 = 1 p q, E XY E X p 1 p E Y q 1 q p = 2 Schwarz ii ( Minkowski p 1, E X + Y p 1 p E X p 1 p + E Y p 1 p iii ( Chebyshev ϕ R (, x >, P ({ω Ω ; X(ω x} E ϕ (X ϕ(x iv ( Jensen ψ R ψ(x, ψ (EX E ψ(x,, 8, pp ( Fubini (S, S, µ = (S 1 S 2, S 1 S 2, µ 1 µ 2 f(z = f(x, y, i f(x, yµ 2 (dy x S 1 -, S 2 f(x, yµ 1 (dx y S 2 - S 1 ii f(zµ(dz = µ 1 (dx f(x, yµ 2 (dy = µ 2 (dy f(x, yµ 1 (dx S S 1 S 2 S 2 S (S, S, µ f(z = f(x, y µ 1 (dx S 1 f(x, y µ 2 (dy, S 2 µ 2 (dy S 2 f(x, y µ 1 (dx, S 1 S f(z µ(dz,, i, ii Fubini

13 ( X 1,, X n N, f 1,, f n R N R ( C f k (X k, n k=1 f k(x k, n n E f k (X k = E f k (X k k=1 13, p88 N X, k=1 µ X (A := P ( {ω Ω ; X(ω A}, A B(R N, µ X ( R N, B(R N X g(x N Borel N X, g(x Eg(X X P X µ X, Eg(X µ X Eg(X = g(x 1,, x N µ X (dx 1,, dx N Lebesgue-Stieltjes ( 8, pp , p83, 1211 X N, µ X X ϕ(ξ := E e 1ξ X = e 1 P N k=1 ξ kx k µ X (dx 1,, dx N, ξ R N X, ξ ξ (, ξ X R N ξ X µ X f ϕ(ξ := E e 1ξ X = e 1 P N k=1 ξ kx k f(x 1,, x N dx 1 dx N, ϕ f Fourier X Y, 1212 ( R N - X, Y µ X, µ Y, ϕ X, ϕ Y, ϕ X = ϕ Y µ X = µ Y 13, p ,

14 X 1, X 2,, X n, ϕ X (ξ 1,, ξ n = n ϕ Xk (ξ k, ξ 1,, ξ n R k=1, ϕ X X = (X 1,, X n 13, p ( 1 N(m, v P (Λ = 1 Λ 2πv e (x m2 /2v dx, Λ B(R 1 ϕ(ξ = e 1ξx 1 e (x m2 2v dx = e 1ξm ξ2 2 v 2πv, Gauss 1215 i X ϕ(ξ m R 1, v R + ϕ(ξ := Ee 1ξX = e 1ξm ξ2 2 v, ξ R 1, X m, v 1 Gauss ii N X = (X 1,, X N ϕ(ξ M R N N N V ϕ(ξ := Ee 1ξ X = e 1ξ M 1 2 ξ V ξ, ξ R N, X M, V N Gauss Gauss X = (X 1,, X N Gauss, N k=1 a kx k Gauss, 9, pp , X =, N k=1 b k(x k X k 1 Gauss (a 1,, a N R N b k = N a i, k = 1,, N i=k, (b 1,, b N R N a k = b k b k+1, k = 1,, N 1, a N = b N

15 {X k }, k = 1, 2, Ω Gauss E X k X 2, k X Gauss 1219 Y, Y 1,, Y n Ω X = (Y, Y 1,, Y n Gauss, Y Y j, j = 1,, n, E(Y EY (Y j EY j =, j = 1,, n Y {Y 1,, Y n } 4, p (Ω, F, P, P (Ω = 1 Ω F, ω Ω, (almost sure, as, X, Y, X = Y as, P ( {ω Ω ; X(ω = Y (ω} = 1 µ,, (almost everywhere µ - ae 131 {X n } n N, X (Ω, F, P i X n X, ({ } P ω Ω ; lim X n (ω = X(ω = 1 n ii, X n X as X n X, ε >, lim P ({ω Ω ; X n(ω X(ω > ε} = n iii, X n X in pr p 1 X n X p (L p, lim E X n X p = n, X n X in L p 132 (Borel - Cantelli Λ n F, n N, P (Λ n < n=1 ( P n=1 k=n Λ k =

16 1 9, k n Λ k n ( ( P Λ k = lim P Λ k lim P (Λ k = n n n=1 k=n k=n, 13, p73, p {X n } n N, X, X (Ω, F, P, i X n X as X n X in pr ii X n X in L p X n X in pr iii X n X in pr X n X in pr k=n X = X as iv X n X in pr X 134 {X n } n N X, i ( X n X ( as lim EX n = EX n ii (Fatou X n E lim inf 13, pp81 82 n X n lim inf n EX n {X n } n N, Lebesgue X, Y 135 {X n } lim sup E X n ; {ω Ω ; X n (ω a} = a, n 136 Y L 1 (P n N X n Y as {X n } 137 {X n } X, i {X n } X ii lim n EX n = EX 138 (Lebesgue X n X as, Y L 1 (P n N X n Y as lim EX n = EX n X n ω, n, Y, 13, pp95 97

17 1 1 14, Radon-Nikodym (Ω, F, P Q : F R i A F, Q(A < ii A n F, n N, A i A j = (i j Q ( n=1 A n = n=1 Q (A n, Q P, P (Λ = Λ F Q(Λ =, Q P 141 (Radon-Nikodym Q (Ω, F, P, Q P, F- Y, Λ F, Q(Λ = Y (ωp (dω (11 Λ,, Y Ỹ (11 Y = Ỹ as Y (Ω, F P Q Radon-Nikodym, Y = dq dp 13, pp , X L 1 (Ω, F, P F σ- G ( G Ω σ- G F Q(A := EX ; A = X(ωP (dω, A G, Q (Ω, G, Q P, Radon-Nikodym, G- Y, Q(A = Y (ωp (dω, A G A 142 Y EX G, G X, 9, pp18 2 A 143 X, Y L 1 (Ω, F, P, G, H F σ- i a, b R EaX + by G = aex G + bey G as ii X Y as EX G EY G as iii iv X G-, Y XY EXY G = XEY G as H G E EX G H = EX H as H = {, Ω} E EX G = EX as

18 1 11 v E X G E X G as vi ( Jensen ψ R ψ(x, ψ (EX G E ψ(x G as vii viii ix f R Borel, f(x X G, E f(x G = E f(x as ( Fatou X n, n N E lim inf n X n G lim inf EX n G n ( Lebesgue X n Y, n N as EY < Y, X n X, n as, lim E X n G = E X G as n Hilbert ( M Hilbert H, H u M M u = u 1 + u 2, u 1 M, u 2 M (, M = {u H ; (u, v =, v M} u 1 u M, u 1 = PM(u 145 M, H 144 H, v M u v = inf u r v = P M(u r M 1, pp X F- X L 2 (P F σ- G, K := {Y L 2 (P ; Y G- } K Hilbert L 2 (P, P K (X = EX G 122, {f n } n N G- lim f n = f f G- n, L 2 (P f L 2 (P K L 2 (P, 144 P K (X Radon-Nikodym, EX G Ω R-, as

19 1 12 i EX G G- ii EX GdP = XdP, A G A A P K (X i, ii P K (X K P K (X G- X P K (X K Y (X P K (XdP =, Y K Ω A G I A (ω K (X P K (XdP = I A (X P K (XdP = P K (XdP = A A Ω A XdP, A G, P K (X = EX G

20 2 Brown 1828 Robert Brown,, Brown {B t (ω} t T, NWiener 1923 Brown,,, Brown,, Brown,, 2 M M 2 M M 21 Brown Ξ N N (Ω, F, P Ξ N - X = {X t (ω} t T = {X t } t T N Ξ-, T, T =, T =, T {X t (ω}, {X t } t T {X t (ω} t T as t,, {X t } t T, {X s } s t σ- σ(x s ; s t Ξ Euclid,, Brown 211 ( Brown (Ω, F, P R N - {B t } t T N Brown, i s < t, B t B s, (t se N Gauss E N N ii iii s < t, B t B s σ(b u ; u s {B t }, B = as 1 Brown {B t } 12, pp5 6 2, pp

21 2 Brown Ω := {ω ; {B t (ω} }, P (Ω = {B t } t T 1, T 2, T 1 < T 2 as 214 ( log i lim sup t B t 2t log(log t = 1 as ii lim inf t B t 2t log(log t = 1 as 215 t >, : = t < t 1 < < t n < t < t n+1, Q t (B; := n Bti+1 t B ti 2, Q (B; = := max t i+1 t i i i= a b := min(a, b t, E (Q t (B; t 2, 216 ( Brown,, R N W N := C(, R N ω 1, ω 2 W N, d(ω 1, ω 2 := n= 1 2 n ( sup ω 1 (t ω 2 (t 1 t n, W N d W N Borel σ- B(W N NWiener (1923, (W N, B(W N P, N Brown {B t (ω} W N ω W N B t (ω = ω(t R N {B t (ω} (W N, B(W N, P N Brown Wiener, Brown Wiener, P Wiener 12, pp7 9, σ- σ- 217 (Ω, F, P F {F t } t T i F s F t F, s < t ii t T F t σ-, {F t } t T (σ-, (Ω, F, P ; F t

22 2 Brown {X t } t T (Ω, F, P F X t := σ(x s ; s t {Ft X } t T {X t } (Ω, F, P ; Ft X, (Ω, F, P ; F t 219 {X t } t T (Ω, F, P ; F t Ξ- i t, X t F t -, {X t } F t ii t, (s, ω, t Ω X s (ω Ξ, B (, t F t, {X t } F t {X t } F t, {X t } F t - t {X t } 123, B (, t F t ( F t -Brown (Ω, F, P ; F t N {B t } i {B t } F t ii s < t, B t B s F s,, (t se N Gauss N F t -Brown ii E e 1ξ (B t B s Fs = e ξ 2 2 (t s, t > s, ξ R N (21, (21, E e 1ξ (B t B s = e ξ 2 2 (t s B t B s N Gauss F s - X, ( ( B t B s Y =, ξ = X ξ 1 ξ 2 ξ 1, ξ 2 R N, E e 1 Y Fs = E e 1ξ1 (B t B s + 1ξ2 X Fs = e 1ξ 2 X E e 1ξ 1 (B t B s Fs = e 1ξ 2 X e ξ (t s

23 2 Brown 16, E e 1 Y = e ξ (t s E e 1ξ2 X = E e 1ξ1 (Bt Bs E e 1ξ2 X, 1213 B t B s X X, B t B s F s 2113 F t := σ ( s<t F s, Ft+ := ε> F t+ε {F t } F t = F t, t, F t = F t+, t {X t } {Ft X }, {X t } {Ft X }, {X t } W N 2, p93,p126 Brown {B t } W N, F t B := σ(b s ; s t, 2114 (Ω, F, P N Brown {B t }, ( Ft B = σ(b s ; s t, F B = σ t F B t N = {F Ω ; F G F B, P (G = } F t = σ(f B t N, F t = F t+, t 12, pp16 17 Brown {B t } {F t } Brown, 2115, (Ω, F, P ; F t τ(ω, t, {ω Ω ; τ(ω t} F t ( stopping time τ t, t, 2116 {F t }, τ(ω, t {ω Ω ; τ(ω < t} F t

24 2 Brown 17 {ω Ω ; τ(ω t} {τ t} ( {τ < t} F t, {τ t} = {τ < t + 1 n } = {τ < t + 1 n }, k N n=1 k n=k {τ < t + 1 } F n t+ 1 k {τ t} k=1 n=k, F t F t+ 1 k = F t+ = F t ( {τ t} F t n {τ t 1 } F n t 1 n {τ < t} = F t, {τ t 1 n } F t n= R- {X t } F t {F t } R G, { inf {t ; X t G}, { } = σ G :=, { } =, σ G {X t } R F σ F ( Q + {σ G t} = {ω ; X s (ω G c, s < t} = {ω ; X r (ω G r<t,r Q c } +, {σ G < t} = {ω ; X r (ω G} = r<t,r Q + r Q + {ω ; X r (ω G, r < t} {ω ; X r (ω G, r < t} F t {σ G < t} F t σ G ( ρ(x, F := inf{ x y ; y F}, G n = {x ; ρ(x, F < 1 } n G n σ Gn, G n G n+1 σ Gn σ Gn+1 {σ Gn }, lim = σ n σ Gn, {σ t} = n=1 {σ G n t} F t σ, {X t } X σ = lim as n F G n σ Gn σ F, n X σg n σ σ F, n X σ G n X σ n=1 G n = F σ F σ σ = σ F, σ σ F 2118 F τ := {A F ; t A {ω Ω ; τ(ω t} F t }, F τ σ-, τ F τ -

25 2 Brown 18 F τ σ- i {τ t} = F t ii iii A F τ A {τ > t} = (A {τ t} {τ > t} F t, A c {τ t} = (A {τ > t} c {τ t} F t A c F τ A n F τ, n = 1, 2, ( n=1 A n {τ t} = n=1 (A n {τ t} F t n=1 A n F τ, A = {ω ; τ(ω a}, a, t a < t A {τ t} = A F a F t, a t A {τ t} = {τ t} F t τ F τ - 12, p (Ω, F, P ; F t σ, τ, i σ τ := max(σ, τ, σ τ := min(σ, τ ii σ τ as F σ F τ iii F σ τ = F σ F τ iv {ω Ω ; τ(ω < σ(ω} F σ τ, {ω Ω ; τ(ω σ(ω} F σ τ (Ω, F, P ; F t {X t } t T, F t -,, i {X t } F t ii iii t T, E X t < t s, t, s T, E X t F s = Xs as, iii E X Fs t Xs as {X t } F t -, E X Fs t Xs as {X t } F t -, 222 X (Ω, F, P ; F t Z t := EX F t {Z t } F t - s t EZ t F s = E EX F t Fs = EX Fs = Z s

26 2 Brown {B t } t T (Ω, F, P ; F t F t -Brown, } i {B t } ii {B 2 σ2 σbt t t} iii {e 2 t, σ F t - i s t, B t B s F s, E B Fs t = Bs E B t B s Fs = E Bt B s = ii E B t 2 B s 2 Fs = E (Bt B s 2 + 2B s (B t B s Fs = E (Bt B s 2 Fs, E B 2 t t Fs = 2 Bs s iii E σ2 σbt e t 2 Fs = E (B t B s 2 = t s σ2 σbs = e 2 t E e σ(bt Bs Fs = e σb s σ2 2 t E e σ(bt Bs = e σb s σ2 2 t e σ2 2 (t s = e σb s σ2 2 s T =,, 12, pp ( Doob {X t } t T (Ω, F, P ; F t, XT = sup X t (ω t T i λ p P ({ω Ω ; XT λ} E X T p, p 1 ii ( p pe XT E XT p p, p > 1 p 1 {X t }, Jensen f {f(x t } {f(x t } {f(x t } 225 ( {X t } t T (Ω, F, P ; F t F t -, σ, τ σ τ K ( as, E X τ Fσ Xσ as, {X t } {X t }, {X t }

27 2 Brown 2 23 (Ω, F, P ; F t {F t } t T, N := {A Ω ; B F st A B, P (B = } F (22 231, A := {A t (ω} t,, i F t ii t 1 < t 2 A t1 A t2 < as A +;c A c := {A 1 A 2 ; A 1, A 2 A +,c }, M c, p 1 M p;c := {M := {M t } t T M c ; t E M t p < }, 232 M M 2,c, M = M A +,c, M 2 M 1 Brown {B t } M 2,c, 223, M M 4,c 233 M M 4,c Q t (M; 215, M = M A +,c, Q t (M; M t in L 2 (Ω, t, M 2 M 12, pp M M c, M A c M t = M as 235 M M 4,c, C, EQ t (M; 2 C 236 {Y (n } n N F t lim E Y (n s Y s (m 2 = n,m sup n s t sup s t, Y F t, ( P lim Y s (n Y s = = 1 lim E Y s (n Y s 2 = n sup s t

28 2 Brown ( M 2 A, M 2 Ã, A, Ã A +,c A Ã = (M 2 Ã (M 2 A, A Ã A c 234, t A t Ãt = A Ã = as ( (Step1, n n, {Q t (M, n } L 2 (Ω Cauchy : t = < t 1 < t 2 < < 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 s < t i < t j E(M tj M ti 2 F s = EM 2 tj M 2 ti F s EQ t (M; F s = k n (M ti+1 s M ti 2 + E (M ti+1 t M ti s 2 Fs i= i=k = Q s (M; + EM t 2 M s 2 F s EQ t (M; Q s (M; F s = EM t 2 M s 2 F s (23 EM t 2 Q t (M; F s = EM t 2 Q s (M; (Q t (M; Q s (M; F s = EM s 2 Q s (M; M 2 Q(M; : t = < t 1 < t 2 < < t m < t < t m+1 <, M 2 Q(M;, 235 Q t (M;, Q t (M; L 2 (Ω L t := Q t (M; Q t (M; {L t } M 2,c (23 EQ t (L; Q (L; F = EL t 2 L 2 F, Q (L; =, L =, EL 2 t = EQ t (L; E (Qt (M; Q t (M; 2 = E Q t ( Q(M; Q(M; ; (24 : s < s 1 < < s l < t < s l+1, t j s k < s k+1 t j+1 Q sk+1 (M; Q sk (M; = (M sk+1 M tj 2 (M sk M tj 2 = (M sk+1 M sk (M sk+1 + M sk 2M tj

29 2 Brown 22 Q t ( Q(M; ; = l ( Qsk+1 t(m; Q sk (M; 2 k= sup k M sk+1 + M sk 2M tj 2 Q t (M; Q t ( Q(M; Q(M; ; 2Q t ( Q(M; ; + 2Q t ( Q(M; ;, (24 Schwarz (Q(M; E Q(M; 2 2E ( Q t Q(M; ; + 2E ( Q t Q(M; ; 1 2E sup M sk+1 + M sk 2M tj 4 2 E Qt (M; k 1 + 2E sup M sk+1 + M sk 2M t j 4 2 E Qt (M; k sup k M sk+1 + M sk 2M tj (Mt 4, Lebesgue lim E sup M sk+1 + M sk 2M tj 4 E lim sup M sk+1 + M sk 2M tj 4, k, M t (ω, t as, 235 lim E(Q(M; Q(M; 2 =,, n {Q t (M; n } L 2 (Ω Cauchy (Step2 t >, n n, n {Q s (M; n Q s (M; m } s t, Doob, m, n E sup Q s (M; n Q s (M; m 2 4E Q t (M; n Q t (M; m 2 s t Q(M; n F t, 236 F t M ( P lim sup Q s (M; n M s = = 1, lim E sup Q s (M; n M s 2 = (25 n s t n s t L 2 (Ω M s L 2 (Ω, s n=1 n, t, n n+1, n, s 1 < s 2 s 1, s 2 n Q s1 (M; n Q s2 (M; n, M s n=1 n M s, t, M s t M A +,c k

30 2 Brown 23 M 2 Q(M; n EM t 2 M t F s EQ t (M; n M t F s = EM t 2 Q t (M; n F s n, 235 (25 = M s 2 M s + ( M s Q s (M; n EQ t (M; n M t F s, M s Q s (M; n as EM 2 t M t F s = EM 2 s M s as M 2 M 232 ρ n = inf {t ; M t n} ρ n, ρ n {M t ρn } := {M ρ n t }, 233 A n A +,c, n (M ρn 2 A n (ρ n+1 t ρ n = ρ n t (M ρ n+1 t ρ n = M ρ n t, ( ρ (M n+1 t 2 A n+1 ρn t = (M ρ n t 2 (A n+1 t ρ n (A n+1 t ρn = A n t, (A m t ρn = A n t, m > n M t := lim (M ρn t 2 M ρn t n A n t = (M ρn t 2 lim m (Am t ρ n = (M ρn t 2 A n t, n E(M ρ n t 2 M ρ n t = EM 2 M =, E M t = lim E M ρ n t = lim E(M ρ n t 2 n n M ρn t 2 Mt 2 L 1 ρn Lebesgue lim E(Mt 2 = EM 2 t < n M t L 1 M 2 t M t L 1, (M ρn 2 M ρn, A F s E(M ρn t 2 M ρn t ; A = E(M ρ n s 2 M ρ n s ; A (M ρn t 2 M ρn t Mt 2 + M t L 1 n Lebesgue EM 2 t M t ; A = EM 2 s M s ; A, M 2 M M, 237 F t {X t } τ n ( t T τ n T, n {Xt τn } := {X t τn } {X t }, M c;loc 233, 232, 12, pp35 36

31 2 Brown M M c,loc, i M = M A +,c, M 2 M M c,loc, M ii iii t >, t { n } ( lim P sup Q s (M; n M s > ε =, ε > n s t M =, M M 2,c, t E M t <, EM 2 t = E M t 239 M, N M c,loc, M, N = M, N A c i MN M, N M c,loc ii t >, t { n } ( lim P sup Q s (M, N; n M, N s > ε =, ε > n s t, : t < t 1 < < t n < s < t n+1 < Q s (M, N; := n (M ti+1 s M ti (N ti+1 s N ti, Q (M, N; = i= Q s (M, N; n = 1{Q 4 s(m + N; n Q s (M N; n }, M, N = 1 ( M + N M N M, N 231 {(Bt 1, Bt 2,, Bt N } N F t -Brown, B i, B j t = δ ij t, δ ij i = j 1, i j B tb i j t δ ij t E(B i tb j t δ ij t (B i sb j s δ ij s F s = E(B i t B i s(b j t B j s + B i s(b j t B j s + B j s(b i t B i s δ ij (t s F s = E(B i t B i s(b j t B j s F s + B i seb j t B j s F s + B j seb i t B i s F s δ ij (t s, E(Bt i Bs(B i j t Bs F j s = δ ij (t s i = j (Bt i Bs i 2 F s E(B i t B i s 2 F s = E(B i t B i s 2 = t s, i j Bt i Bs i Bj t Bs j F s E(B i t B i s(b j t B j s F s = EB i t B i seb j t B j s =

32 2 Brown 25 E(B i tb j t δ ij t (B i sb j s δ ij s F s = BtB i j t δ ij t B i, B j t = δ ij t M, N 12, p M, N, L M c,loc, i M, N = N, M ii M + N, L = M, L + N, L iii iv a am, N = a N, M τ M τ, N τ t = M τ, N t = M, N t τ

33 3 2 M, M Brown, (1942 M, (1967, 2 Hilbert, f M M,, Brown Lévy 31 (Ω, F, P ; F t, {F t } t T, (22 M 2;c (, T := { } M ; EM 2 T <, {M t } t,t F t - { T } L 2 (, T ; M := ϕ ; ϕ F t - E ϕ(s, ω 2 d M s < { T } L 2 ( M := ϕ ; ϕ F t - E ϕ(s, ω 2 d M s <, T, M M 2 (, T, M T := EM T M 2,c (, T T Hilbert, (M, N = EM T N T {M (n t } n N M 2,c (, T, T } T Cauchy Doob, E M (n t M (m t 2 M (n 4E T M (m T 2 sup t T, lim E n,m sup t T {M (n M (n t M (m 2 = t 26

34 3 27, {M (n t } F t, 236 F t {M t }, lim M (n 2 t M t = (31 n E sup t T lim M (n M T = {M t } n, (31 t, T t } L 2 (Ω, F, P Cauchy, L 2, M t L 2, Jensen, A F s, s t n EM (n s {M (n M s ; A, EM (n t M t ; A, {M (n } EM (n ; A = EM (n ; A, t EM s ; A = EM t ; A s {M t } {M t } M 2,c (, T, M 2,c (, T M, N M 2,c (, T EM T N T, EM T M T 1 2 = M T M 2,c (, T T Hilbert 312 M M 2,c, ϕ L 2 (, T ; M, T ϕ T := E ϕ(s, ω 2 d M s 1 2, T L 2 (, T ; M, ϕ T, ϕ T = ϕ = aϕ T = a ϕ T (a R ϕ, ψ L 2 (, T ; M, Lebesgue-Stieltjes d M Minkowski ( T 1 ( (ϕ + ψ 2 2 T 1 ( d M s ϕ 2 2 T 1 d M s + ψ 2 2 d M s, ϕ + ψ dp Minkowski ϕ + ψ T E( ϕ + ψ E ϕ E ψ2 1 2 = ϕ T + ψ T ϕ (, ϕ ( 313 : = t < t 1 < < t k+1, k = 1, 2,, F ti - ξ i (ω k ϕ ( (t, ω = ξ i (ωi (ti,t i+1 (t ϕ (, L i=1 t

35 ϕ L 2 ( M, ϕ ( L, n N, n ϕ ( n ϕ T, n, L L 2 ( M ϕ L 2 ( M, ϕ (m (t, ω := ϕ(t, ωi m,m (ϕ(t, ω, m N, ϕ (m L 2 ( M, (t, ω, lim m ϕ (m (t, ω ϕ(t, ω = ϕ (m (t, ω ϕ(t, ω 2 2 ( ϕ (m (t, ω 2 + ϕ(t, ω 2 4 ϕ(t, ω 2, ϕ(t, ω 2 Lebesgue lim m ϕ (m (t, ω ϕ(t, ω T =, ϕ, ϕ n (s, ω := s (s 1 n ϕ(u, ω d M u M s M (s 1 n, ϕ(s, ω := lim sup n ϕ n (s, ω, ϕ n, ϕ ϕ, T ϕ d M s <, T F (s := s ϕ(u, ω d M u, df (s, df (s d M s as Radon- Nikodym, as df (s/d M s lim ϕ n(s, ω = n df (s d M s (ω d M s - ae ( 8, p139 lim ϕ n(s, ω = n, t T df (s d M s (ω = ϕ(s, ω (32 ϕ(s, ω d M s = = I,T Ω (s, ω df (s I,T Ω (s, ω df (s d M s d M s = ϕ(s, ω d M s (33

36 3 29 ϕ {ϕ n (s, ω} F t, : = t < t 1 < < t k k ϕ ( n (s, ω := ϕ n (t i, ωi (ti,t i+1 (s i= L ϕ n ϕ ( n lim ϕ( n (s, ω = ϕ n (s, ω, (s, ω, T lim E ϕ ( n (s, ω ϕ n (s, ω 2 d M s = (32 T lim E ϕ n (s, ω ϕ(s, ω 2 d M s = n, (33 Radon-Nikodym ϕ(s, ω = ϕ(s, ω d M s - ae T E ϕ(s, ω ϕ(s, ω 2 d M s = n, T E ϕ ( n ϕ 2 d M s 5 { T E ϕ ( n ϕ n 2 d M s T T } + E ϕ n ϕ 2 d M s + E ϕ ϕ 2 d M s 315 ( M M 2,c ϕ ( (t, ω = n i= ξ i(ωi (ti,t i+1 (t L, I(ϕ ( (t, ω := n ξ i (ω ( M ti+1 t M ti t, t, T i=, I(ϕ ( (t, ω = ϕ ( (s, ω dm s 316 I(ϕ ( (t, ω i I(ϕ ( M 2,c ii I(ϕ ( t = ϕ( (s, ω 2 d M s

37 3 3 iii E I(ϕ ( 2 t t = E ϕ( (s, ω 2 d M s s < t s, t R : = t < t 1 <, t k s < t k+1 < t l t < t l+1, k, l N i m < k Eξ m (M tm+1 M tm F s = ξ m (M tm+1 M tm m = k Eξ m (M tm+1 M tm F s = ξ k E(M tk+1 M s + (M s M tk F s = ξ k (M s M tk m > k Eξ m (M tm+1 M tm F s = E Eξ m (M tm+1 M tm F tm Fs = EI(ϕ ( (t, ω F s = I(ϕ ( (s, ω I(ϕ (, ξ i, M L 2 (Ω I(ϕ ( L 2 (Ω, M ti t as I(ϕ ( (, ω as I(ϕ ( M 2,c ii s t i < t j F s F ti F tj E ξ i (M ti+1 M ti ξ j (M tj+1 M tj F s = E Eξi ξ j (M ti+1 M ti (M tj+1 M tj F tj Fs = E ξ i ξ j (M ti+1 M ti EM tj+1 M tj F tj Fs = E { l } 2 {I(ϕ ( (t, ω I(ϕ ( (s, ω} 2 F s = E Fs ξ i (M ti+1 t M ti s s t m F s F tm E ξ 2 m (M tm+1 M tm 2 F s = i=k l E ξ 2 i (M ti+1 t M ti s 2 F s i=k M 2 M = E Eξ m 2 (M tm+1 M tm 2 F tm Fs = E ξm 2 EM tm+1 2 M tm 2 F tm Fs = E ξ m 2 E(M tm+1 2 M tm+1 (M tm 2 M tm + ( M tm+1 M tm F tm Fs = E ξ m 2 ( M tm+1 M tm F s E {I(ϕ ( (t, ω I(ϕ ( (s, ω} 2 F s = I(ϕ ( M 2,c, l E ξ 2 i ( M ti+1 t M ti s F s i=k = E ϕ ( (r, ω 2 d M Fs r s = E ϕ ( (r, ω 2 d M Fs r s ϕ ( (r, ω 2 d M r (34 E {I(ϕ ( (t, ω I(ϕ ( (s, ω} 2 F s = E I(ϕ ( (t, ω 2 I(ϕ ( (s, ω 2 F s = E I(ϕ ( (t, ω 2 F s I(ϕ ( (s, ω 2 (35

38 3 31 (34, (35 { I(ϕ ( (t, ω 2 ϕ( (s, ω 2 } d M s, ii iii E I(ϕ ( (t, ω 2 ϕ ( (s, ω 2 d M s = E I(ϕ ( (, ω 2 ϕ ( (s, ω 2 d M s = iii f L 2 ( M 317 ( 314, f L 2 ( M f n L T lim E f n (s, ω f(s, ω 2 d M s =, T (36 n X (n t := f n(s, ωdm s, t lim E sup X s (n X s 2 = n s t {X t } M 2,c X t f M M 2,c, X t = f(s, ω dm s, {X t }, 316 {X (n } M 2,c, n, m X (n t {X (n t {X (n E X (n t X (m t E X (m t 2 = E f n (s f m (s 2 d M s, t t } M 2,c Doob, n, m sup X s (n X s (m 2 4E X (n T X (m T 2, T s T t } t,t M 2,c (, T, T Cauchy , {Xt T } t,t M 2,c (, T ( lim E sup X (n t Xt T 2 = P lim sup X (n t Xt T = = 1 (37 n t T n t T t T 1 T 2, n E sup X T 1 t X T 2 t 2 2E sup X T 1 t X (n t 2 + 2E sup X T 2 t X (n t 2 t T 1 t T 1 t T 2, Xt T T X t := lim T XT t {X t } M 2,c, (37 sup t X (n X t t

39 3 32 X t f n f n, fn L (36, {X t }, { X t } f n, fn, Doob E sup X s X s 2 4E X t X t 2 s t { 12 E X t X (n t 2 + E X (n (m t X t 2 (m + E X t X } t 2 n, m, 236, E X (n t E X t X (n t 2 (m, E X t X t 2 (m X t 2 E 2{ f n (s f(s 2 + f(s f n (s 2 } d M s Esup s t X s X s 2 =, t X = X as 318 X, M, f 317 EX 2 t = E M t = E f(s, ω 2 d M s { X 2 t X (n t f(s, } ω2 d M s 317 A F s t ( E X 2 t f(r, ω d M r X (n 2 t f n (r, ω d M r ; A Xt E 2 X (n 2 t t ; A + E (f f n d M r ; A (38 Hölder Xt E 2 X (n 2 t ; A E X t X (n t ( X t + X (n t ; A ( E X t X (n 1 t 2 2 ; A E X t 2 ; A E X (n t E X t 2 ; A, E X (n t 2 ; A < n (37 Xt E 2 X (n 2 t ; A (36 t t E (f f n d M r ; A E f f n d M r 2 ; A { (38 n as 2 Xt f(s, } ω2 d M s, I : L 2 (, T ; M f X T = T f(s, ω dm s M 2,c (, T T, f T = X T 1 2

40 Schwarz, 321 ( M, N M 2,c, ϕ(s, ω, ψ(s, ω t > ϕ(s, ω 2 d M s <, ψ(s, ω 2 d N s < t, λ t (ω := 1 2 ( t ϕ(s, ωψ(s, ω d M, N s ϕ(s, ω 2 d M s 1 2 ( ψ(s, ω 2 d N s 1 2 as M t + N t (ω u, t, λ t (ω λ u (ω = 1 2 ( M t M u + ( N t N u = M t M u = as, N t N u = as M t, N t λ t (ω as 2311, r R Ω r Ω P (Ω r = 1, ω Ω r M + rn = M + 2r M, N + r 2 N Ω = r Q Ω r P (Ω = 1, ω Ω u d M + rn s = u d M s + 2r u d M, N s + r 2 d N s (39, r, (39 r R as u 2 ( ( d M, N s d M s d N s u λ t (ω λ u (ω = M, N t M, N u = as M, N t λ t (ω as Ω 1 Ω, P (Ω 1 = 1 Ω 1 M t, N t, M, N t λ t (ω Radon-Nikodym M t (ω = f 1 (s, ω dλ s (ω, N t (ω = M, N t (ω = f 1, f 2, f 3 u f 3 (s, ω dλ s (ω u f 2 (s, ω dλ s (ω

41 , α, β R, P (Ω αβ = 1 Ω αβ Ω 1, ω Ω αβ αm + βn t2 (ω αm + βn t1 (ω 2 ( = α 2 f 1 (s, ω + 2αβf 3 (s, ω + β 2 f 2 (s, ω dλ s (ω t 1 t 1 < t 2 < t 1, t 2 R ω Ω αβ dλ T αβ (ω t(ω = T αβ (ω B(,, t / T αβ (ω α 2 f 1 (t, ω + 2αβf 3 (t, ω + β 2 f 2 (t, ω (31 Ω := α,β Q Ω αβ, ω Ω T (ω := α,β Q T αβ(ω, P ( Ω = 1 dλ T (ω t(ω =, (31 t / T (ω (α, β, (31 α, β, t / T (ω (α, β α = αϕ(t, ω, β = ψ(t, ω t / T (ω α 2 ϕ(t, ω 2 f 1 (t, ω + 2αϕ(t, ωψ(t, ωf 3 (t, ω + ψ(t, ω 2 f 2 (t, ω dλ t (ω, α R t, α 2 ϕ(s, ω 2 d M s + 2α 2 ( ϕ(s, ωψ(s, ω d M, N s ϕ(s, ωψ(s, ω d M, N s + ψ(s, ω 2 d N s ( ϕ(s, ω 2 d M s ψ(s, ω 2 d N s 322 M M 2,c, f(s, ω L 2 ( M, X t = f(s, ω dm s N M 2,c X, N t = f(s, ω d M, N s (311, (311 X M 2,c X = 322, f(s, ω dm s (311 Lebesgue-Stieltjes, f L 2 ( M f n L, lim E f(s, ω f n (s, ω 2 dm s =, t n

42 3 35 X (n t = f n(s, ω dm s 318 E X (n X t = E f(s, ω f n (s, ω 2 dm s, n Hölder E X (n, N t X, N t = E X (n X, N t t ( t = E d X (n X, N s E d X (n X s 1 2 ( 1 2 d N s = E X (n X 1 2 t N 1 2 t E X (n X t 1 2 E N t 1 2, n (312 t E f(s, ω d M, N s f n (s, ω d M, N s E f(s, ω f n (s, ω 2 d M s 1 2 E N t 1 2, n (313 { X (n t N t f } n(s, ω d M, N s t k s < t k+1 t l t < t l+1, f n (s, ω = i ξ i(ωi (ti,t i+1 (s EX (n t N t X s (n N s F s { l } { k } Fs = E ξ i (M ti+1 t M ti s N t + E ξ j (M tj+1 s M tj (N t N s F s i=k N, M, N, MN M, N, i k + 1 Eξ i (M ti+1 M ti N t F s = E Eξ i (M ti+1 M ti N t F ti+1 F s j= = Eξ i M ti+1 N ti+1 F s E Eξ i M ti N ti+1 F ti Fs = Eξ i (M ti+1 N ti+1 M ti N ti F s = Eξ i ( M, N ti+1 M, N ti F s = E EX (n t N t X s (n N s F s = E i+1 Fs f n (r, ω d M, N r s Fs = E f n (r, ω d M, N r t i f n (s, ω d M, N s F s s f n (r, ω d M, N r { (n X t N t f } n(s, ω d M, N s X (n, N t = f n (s, ω d M, N s

43 3 36 (313 t E f(s, ω d M, N s X (n, N t, n (314 (312, (314 X (n, N X, N in pr X (n, N f(s, ω d M, N s in pr 133 X, N t = f(s, ω d M, N s as X, N t = f(s, ω d M, N s, N M 2,c X, X X, N = X, N X, N =, N = X X X X = as (X X 2 E(X t X t 2 =, t X = X as 323 i M M 2,c, f, g L 2 ( M, a, b R (af(s, ω + bg(s, ω dm s = a f(s, ω dm s + b g(s, ω dm s, t ii M, N M 2,c, f L 2 ( M L 2 ( N, f L 2 ( am + bn f(s, ω d(am + bn s = a f(s, ω dm s + b f(s, ω dn s, t iii M M 2,c, f L 2 ( M, N = f(s, ω dm s, g L 2 ( N, fg L 2 ( M g(s, ω dn s = f(s, ωg(s, ω dm s, t, i (af + bg dm = af dm + bg dm ii f d(am + bn = af dm + bf dn iii dn = f dm g dn = fg dm, i N M 2,c, (af + bg dm, N = (af + bg d M, N = a f d M, N + b g d M, N = a f dm, N + b g dm, N = a f dm + b g dm, N,

44 3 37 ii 2311 f 2 d am + bn = f 2 d(a 2 M + 2ab M, N + b 2 N a 2 ( f 2 d M + 2ab 1 ( 1 f 2 2 d M f 2 2 d N + b 2 f 2 d N E f(s, ω2 d am + bn s <, t f L 2 ( am + bn, L M 2,c f d(am + bn, L = a f dm + b f dn, L iii 322 N t = f(s, ω2 d M s E f(s, ω 2 g(s, ω 2 d M s = E g(s, ω 2 d N s <, t fg L 2 ( M, 322 L M 2,c N, L = f d M, L fg dm, L = fg d M, L = g d N, L = g dn, L, ( M M c,loc f(s, ω ( T P f(s, ω 2 d M s < = 1, T I(f(t, ω := f(s, ω dm s M M c,loc σ n as M t σn M 2,c { } τ n (ω := n inf t ; f(s, ω 2 d M s n, 2117 τ n τ n as ρ n = τ n σ n, M (n t = M t ρn, f (n (t, ω = f(t, ωi,ρn (t, {M (n t } } M 2,c, f (n L 2 ( M (n {M (n t I(f (n (t, ω := f (n (s, ω dm (n s = ρn f(s, ω dm s I(f (n M 2,c n ρ n I(f(t, ω := I(f (n (t, ω, t ρ n I(f M f

45 3 38, t < ρ n, n m I(f (n (t, ω = I(f (m (t, ω, N M 2,c I(f (m (t ρ n, N t = = ρn ρn ρm f (m dm, N t f dm, N t = I(f (m (t ρ n = I(f (n (t ρn f dm, N t = I(f (n, N t 325 M M c,loc, f(s, ω P ( T f(s, ω2 d M s < = 1, T {X t } = { f(s, ω dm s} Mc,loc, X = N M c,loc X, N t = f(s, ω d M, N s, t as σ n, τ n, ρ n, f (n 324, X (n t = ρ n f(s, ω dm s N M c,loc τ n = inf {t ; N t n}, η n = ρ n τ n, η n ρ n X, N t ηn = X, N η n t = X ρ n, N η n t = = f (n (s, ω dms ρn, N ηn t = f (n (s, ω d M, N s ηn = n η n X, N t = ηn f(s, ω d M, N s f (n (s, ω d M ρn, N ηn s f(s, ω d M, N s, X, N t = f(s, ω d M, N s, N M c,loc X X X, N =, N M c,loc, N = X X X X = as X = X as 33 (,,, 331 X F -, M M c,loc, A A c X = {X t } X t = X + M t + A t, M = as, A = as

46 3 39, {X t } R N -,, C k (R N R N k 332 ( X t = (Xt 1, Xt 2,, Xt N, X i t = X i + M i t + A i t, i = 1, 2,, N N, f C 2 (R N, N N f(x t = f(x + D i f(x s dms i + D i f(x s da i s i= N i,j=1, D i f := f, D x i ij f := 2 f x i x j i=1 D ij f(x s d M i, M j s as (315, i, j = 1, 2,, N, N X f C 2 (R N, f(x M M c,loc, f, g C 2 (R 1, df(g(t = f (g(t dg(t, d(f(m t = f (M t dm t f (M t d M t, 2 Taylor 23 {τ n } n N { inf {t ; X > n or M t > n or A t > n or N i=1 τ n := M i t > n}, { } =, { } = 2117 τ n, τ n as, {X t τn } (315,, τ n > t, X, A t, M t, M i, M j t, X L, sup t T A t L, sup t T M t L (L, X t 3L f C 2 (R N 3L, 3L N f, D i f, D ij f, K > f + i D if + i,j D ijf K t, T,, t : = t < t 1 < < t n = t Taylor, n 1 ( f(x t f(x = f(xtk+1 f(x t k = n 1 k= k= N D i f(x tk (X itk+1 X itk i=1 n 1 N k= i,j=1 D ij f(ξ k (X itk+1 X (X itk jtk+1 X jtk

47 3 4 (, ξ k = X tk + θ k Xtk+1 X tk, θk 1 I ( 1 := I ( 2 := n 1 k= n 1 k= N i=1 N i=1 I ( 3 := 1 n 1 2 I ( 4 := n 1 k= i,j=1 k= i,j=1 I ( 5 := 1 n 1 2 D i f(x tk (A itk+1 A itk D i f(x tk (M itk+1 M itk N D ij f(ξ k (A itk+1 A (A itk jtk+1 A jtk N D ij f(ξ k (A itk+1 A (M itk jtk+1 M jtk N k= i,j=1, (1 I ( 1 I 1 := (2 I ( 2 I 2 := D ij f(ξ k (M itk+1 M (M itk jtk+1 M jtk N i=1 N i=1 D i f(x s da i s D i f(x s dm i s as in pr (3 I ( 3 as, I ( 4 as (4 I ( 5 I 5 := 1 N D ij f(x s d M i, M j in L 1 2 i,j=1, 133 t f(x t f(x = 5 j=1 I ( j I 1 + I 2 + I 5 in pr, 133 f(x t f(x = I 1 + I 2 + I 5 as (1 A i t(ω ω t, I ( 1 Lebesgue-Stieltjes I 1 (2 fi (s := n 1 k= D if(x tk I (tk,t k+1 (s, fi ( := D i f(x, D i f(x fi (s X i( t := f i (s dm i s, X i t := D i f(x s dm i s D i f, s, t, fi (s D i f(x s, fi t (s D i f(x s, f i (s D i f(x s 2 d M s, Lebesgue, T E fi (s D i f(x s 2 d M s, as

48 , Esup t T X i( t Xt i 2, X i( t Xt i in L 2, 133 I ( 2 I 2 in pr (3 A j = A j+ A j, A j+, A j A +,c, A j+, A j A j+, A j A j t A j+, A j, M j t I K sup N ( A j t k+1 A j t k A j+ + A j, as j,k j=1 N ( I 4 K sup M j t k+1 M j t k A j+ + A j, as j,k (4, I ( 5 := 1 n 1 2 I ( 5 I ( 5 1 n 1 2 N k= i,j=1 N k= i,j=1 l n sup i,j,l j=1 D ij f(x tk (M itk+1 M (M itk jtk+1 M jtk sup, Q t (M m ; = n 1 D ij f(ξ l D ij f(x tl k= D ij f(ξ l D ij f(x tl Mt i k+1 Mt i M j k t k+1 M j t k N n,m=1 Schwarz, Q t (M n ; 1 2 Qt (M m ; 1 2 ( 2 Mt m k+1 Mt m k, m = 1, 2,, N I ( E 5 I ( 5 1 N E sup D ij f(ξ k D ij f(x tk Q t (M n ; 1 2 Qt (M m ; n,m=1 i,j,k 1 N 1 E sup D ij f(ξ k D ij f(x tk 2 2 E Qt (M n ; Q t (M m ; n,m=1 i,j,k 1 ( 1 sup 2 E D ij f(ξ k D ij f(x tk 2 2 N E Q t (M n ; E Q t (M m ; i,j,k n,m=1 M, 235, N n,m=1 E Q t (M n ; E Q t (M m ; <, D ij f X, D ij f(ξ k D ij f(x tk 2,, E I ( 5 I ( 5, I ( 5 := 1 n 1 2 k= i,j=1 N D ij f(x tk ( M i, M j tk+1 M i, M j tk

49 3 42, I ( E 5 I ( 2 5 ( = 1 n 1 N ( k E D ij f(x tk (Mt i k+1 Mt i k (M j t k+1 M j t k d M i, M j s t k := 1 ( n 1 4 E k= i,j=1 N k= i,j=1 2 V ij k M i, M j, M i M j M i, M j, E (M i t M i s(m j t M j s ( M i, M j t M i, M j s F s =, t > s, k < l E V ij k V pq l = E E V ij = E k V pq l V ij k D pqf(x tl E Ftl (M p t l+1 M p t l (M q t l+1 M q t l l+1 t l d M p, M q s F tl =, (3 M i, M j = M i, M j + M i, M j 1 ( n 1 4 E N k= i,j=1 ( 1 n 1 N 4 K2 E k= n 1 N K 2 E K 2 E k= sup i,k i,j=1 2 V ij k = 1 n 1 4 i,j=1 i,j=1 k= ( N E i,j=1 2 V ij k ( (M i t k+1 M i t k (M j t k+1 M j t k ( M i t k+1 M i t k 2 ( M j t k+1 M j t k 2 + N N Mt i k+1 Mt i k 2 Q t (M j ; j=1 k+1 i,j=1 t k (k+1 N ( +K 2 E sup M i, M j tk+1 M i, M j tk M i, M j + t k d M i, M j s t k 2 d M i, M j s + M i, M j t 2, Hölder I ( E 5 I ( K 2 E sup Mt i k+1 Mt i k 4 2 ( N E Q t (M j ; i,k j=1 N 1 +K 2 E sup M i, M j tk+1 M i, M j 2 tk 2 ( M E i, M j + t + M i, M j t k i,j=

50 E( N j=1 Q t(m j ; 2 < C 1, M i +M j M i M j E( M i, M j + t + M i, M j t 2 < C 2, M i, M i, M j I ( E 5 I ( 2 5,, I ( N i,j=1 D ij f(x s d M i, M j s, as (316, I ( 5, N i,j=1 D ijf(x s d M i, M j s (316 L 1 I ( N i,j=1 D ij f(x s d M i, M j s, in L 1 (315 t as, t, t, t, (315 1 t 333 M M c,loc A, Ã A c M, A t = A, M t = A, Ã t = 315 (3, M 1,, M N M c,loc, M = (M 1,, M N M c,loc 334 f(x, t : R, C x 2, t 1 f f 2 f =, M M c,loc f(m, M M c,loc + 1 t 2 x 2 M, t M, M s = M, M s =, s, t, f(m t, M t f(m, M f = x (M s, M s dm s + f t (M s, M s d M s f 2 x (M s, M 2 s d M s, f(m t, M t f(m, M = f (M x s, M s dm s f(m, M M c,loc Brown, 334 λ2 λx f(x, t = e 2 t, λ C f 334 B = (B 1,, B N

51 3 44 N F t -Brown, M t := ξ B t = N i=1 ξ ibt, i ξ R N, M t = ξ 2 t M t 2 ξ 2 t = (ξ (B t B s + ξ B s 2 ξ 2 t = i,j ξ i ξ j (B i t B i s(b j t B j s + 2 i,j ξ i ξ j (B i t B i sb j s + i,j ξ i ξ j B i sb j s ξ 2 t, 231 E(Bt i Bs(B i j t Bs F j s = δ ij (t s EM t 2 ξ 2 t F s = i ξ i 2 (t s + i,j ξ i ξ j B i sb j s ξ 2 t = M s 2 ξ 2 s, {M 2 t ξ 2 t} f(x, t = e 1x+ 1 2 t 334, { e 1ξ B t + ξ 2 t} 2 := {X t } σ n {X t σn }, e 1ξ B t + ξ 2 t 2 e 1 2 ξ 2t {X t } 223 E e 1ξ B t + ξ 2 t 2 Fs = e 1ξ B s + ξ 2 2 s, E e 1ξ B t + ξ 2 2 t Fs, E = E e 1ξ (B t B s e 1ξ B s e ξ 2 2 t Fs = e 1ξ B s+ ξ 2 2 t E e 1ξ (B t B s F s e 1ξ (B t B s Fs = e ξ 2 2 (t s, s < t 2112 F t -Brown Brown, ( Lévy X = (X 1,, X N, F t - X i, X j t = δ ij t, X N F t -Brown X i t = t < X i M 2,c, X M 2,c ξ R N N ξ X t = ξ i Xt i = i=1 N ξ i ξ j X i, X j t = ξ 2 t i,j, t 335 E e 1ξ (X t X s Fs = e ξ 2 2 (t s, s < t e 1ξ X t + ξ 2 X F t -Brown

52 4 (,,,, 41 (Ω, F, P ; F t, {F t } t (22 {B t } t F t -Brown, N m σ R N - b σ = σ(t, x, ω :, T R N Ω R N m b = b(t, x, ω :, T R N Ω R N, B(, T B(R N F T -, x F t - ξ(ω F - R N -, {X t } t X t = ξ + σ(s, X s, ω db s + b(s, X s, ω ds (41 (41 dx t = σ(t, X t, ω db t + b(t, X t, ω dt, X = ξ (41, B s = s, (41, m Xt i = ξ i + σj(s, i X s, ω dbs j + b i (s, X s, ω ds, i = 1,, N (42 j=1 411 X t (ω :, T Ω R N (41, X = {X t (ω} t T F t, i σ(t, X t, ω, b(t, X t, ω F t -, ( P { σ(t, X t, ω 2 + b(t, X t, ω } dt < = 1, σ 2 := m N j=1 i=1 σi j 2 = tr(σσ 45

53 4 46 ii t T 1 (41 i (41 σ, b ω (t, x, (41 Markov 412 T > t, T, x, y R N ω Ω, K := K T >, σ(t, x, ω, b(t, x, ω { σ(t, x, ω σ(t, y, ω K x y b(t, x, ω b(t, y, ω K x y (43 σ(t, x, ω + b(t, x, ω K(1 + x (44 E ξ 2 <,, T (41, E sup X t 2 < (45 t T, X X, ({ } P ω ; sup t T X t (ω X t(ω > = (Step3 413 ( Gronwall f(t, t, T f(t a + b f(s ds, a, b, f(t ae bt, t, T u(t = e bt t f(s ds, u (t = e bt (f(t b f(s ds ae bt, u(t ae bs ds = (a/b(1 e bt, 412 (Step1 X (n = {X (n t } t,t, n N n =1 X (1 t = ξ n 2 X (n t = GX (n 1 := ξ + σ(s, X (n 1 s, ω db s + b(s, X (n 1 s, ω ds (46, n N, X (n i, ii

54 4 47 i X (n t ii F t Esup t T X (n t 2 < (44 E σ(t, X (n 1 t, ω 2 K 2 E(1 + X (n 1 t 2 <, t, T, E T σ(t, X(n 1 t, ω 2 dt < σ L 2 (, T ; B, T b(t, X(n 1 t, ω dt < as, σ, b (46 {X (n }, n N i, ii n = 1 i, ii X (n 1 t (46, X (n t F t i ii (a + b + c 2 3(a 2 + b 2 + c 2, E sup X (n t 2 t T 2 2 3E ξ 2 + 3E σ(s, X s (n 1, ω db s + 3E b(s, X s (n 1, ω ds sup t T sup t T Doob, Schwarz, (44, E sup X (n t 2 t T T 3E ξ E σ(s, X s (n 1, ω db s + 3E sup t b(s, X (n 1 s, ω 2 ds t T T 3E ξ E σ(s, X (n 1 s, ω T 2 ds + 3T E b(s, X (n 1 s, ω 2 ds T ( 2 3E ξ 2 + (12 + 3T K E sup X s (n 1 ds < (47 s T (Step2 {X (n } n, n 3 t, T, E sup X (n r X r (n 1 2 r { 2E sup σ(s, X (n 1 s, ω σ(s, X s (n 2, ω } 2 db s r t r t r { +2E sup b(s, X (n 1 s, ω b(s, X s (n 2, ω } 2 ds r t 8E σ(s, X (n 1 s, ω σ(s, X s (n 2, ω 2 ds +2tE b(s, X (n 1 s, ω b(s, X s (n 2, ω 2 ds C 1 E X (n 1 X (n 2 2 ds (48 sup r s r r

55 4 48, C 1 := (8 + 2T K 2 2 Doob Schwarz, (43 Fubini n 2, T E X (n r X r (n 1 2 C 1 E sup X (n 1 r X r (n 2 2 ds 1 r s 1 sup r T T s1 n 2 C 1 = C 2 (C 1 T n 2 (n 2! sn 3 E sup r T r X r (1 2 ds n 2 ds n 3 ds 1 X (2, C 2 := E sup r T X r (2 X r (1 2 < (49 Chebyshev, ({ P ω Ω ; X (n (ω X (n 1 (ω > 1 } 2 n, P n=2 sup t T t ({ ω Ω ; sup t T t X (n t 16C 2 (4C 1 T n 2 (n 2! (ω X (n 1 (ω > 1 } < 2 n Borel-Cantelli, ( { P ω Ω ; sup X (k t (ω X (k 1 t (ω > 1 } = t T 2 k n=2 k=n C(R N ( R N sup t T X t (ω X (n t (ω 1, T Fatou (47, E sup X t 2 lim inf E X (n t 2 < t T n, (45, m < n (49, E sup t T X (m t X (n 2 t t sup t T n i=m+1 C 2 (C 1 T i 2 (i 2! (49, Fatou E sup t T X (m 2 t X t lim inf n n i=m+1 C 2 (C 1 T i 2 (i 2! (41

56 4 49 lim sup E m sup t T X (m 2 t X t i=2 C 2 (C 1 T i 2 (i 2! lim sup m m i=2 C 2 (C 1 T i 2 (i 2! = Esup t T X (n t X t 2, n, Doob Schwarz (43, Fubini, (48 T E sup GX (n t GX t 2 C 1 E X s (n X s 2 ds t T (41 Esup t T X (n t T lim E sup GX (n t GX t 2 C 1 n t T, E sup X t GX t 2 t T { lim E sup X t X (n t 2 + E n t T = lim E X (n t GX (n t 2 = lim n sup t T = lim n C 2 (C 1 T n 1 (n 1! = sup t T n E X t 2 C 2 e C1T <, X (n t sup t T lim n E X(n s X s 2 ds = GX (n 2 + E t X (n t X (n+1 t 2 sup t T GX (n t GX t 2 } P (sup t T X t GX t = = 1, T X t = GX t as {X t }, T (41 (Step3 X, X (41 n N { inf {t T ; X t X τ n := t n}, { } = T, { } =, 2117 τ n, (45 lim P (τ n < T = n (48, E sup r t X r τn X r τ 2 n 2E sup r t +2E C 1 E r τn sup r t sup r s {σ(s, X s, ω σ(s, X s, ω} db s 2 r τn {b(s, X s, ω b(s, X s, ω} ds ds Xr τn X r τ n 2 2

57 4 5, 413, E sup t T X t τn X t τ n 2 =, P Xt τn X t τ n > = sup t T Fatou lim inf P P n lim inf n sup t T sup t T Xt τn X t τ n > Xt τn X t τ n > = P sup X t X t > t T σ, a, b R dx t = σ db t + (ax t + b dt, X = ξ X t = e at (ξ + b e as ds + σ e as db s {(X t, t} f(x, t = e at x e at X t e X = = = σ f x (s, X sσ db s + e as σ db s + e as db s + b f x (s, X s(ax s + b ds + e as (ax s + b ds e as ds ae as X s ds f t (s, X s ds Markov X t = ξ + σ(s db s + (A(sX s + b(s ds (411, σ(s :, T R N m, A(s :, T R N N, b(s :, T R N, ξ : Ω R N F - E ξ 2 < ( , X = {X t } t T dy (t dt = A(tY (t + b(t, Y ( = ξ (412

58 4 51 Y (t dφ(t dt = A(tΦ(t, Φ( = E (413 Φ(t Y (t = Φ(t ( ξ + Φ 1 (sb(s ds (414, E 422 (411, (413 Φ(t ( X t = Φ(t ξ + Φ 1 (sb(s ds + Φ 1 (sσ(s db s (415 Z(t := Φ(t Φ 1 (sσ(sdb s, {(Y (t, Z(t} f(y, z = y + z Y (t + Z(t ξ =, (412 (413 (414 dy (s = (A(sY (s + b(s ds { s } dz(s = d Φ(s Φ 1 (rσ(r db r = A(sΦ(s ds s Y (t + Z(t = ξ + 1 dy (s + = dφ(s ds ds s 1 dz(s Φ 1 (rσ(r db r + Φ(sΦ 1 (sσ(s db s Φ 1 (rσ(r db r + σ(s db s = A(sZ(s ds + σ(s db s A(s(Y (s + Z(s ds + b(s ds + σ(s db s, X t = Y (t + Z(t 423 (Ornstein-Uhlenbeck Brown N = m = 1, A(s = α <, b(s =, σ(s = σ > (411 (193 dx t = αx t dt + σ db t Langevin, Brown (415, X t = X e αt + σ e α(t s db s

59 4 52 {X t } Ornstein-Uhlenbeck Brown X t m(t, Cov(X s, X t, V (t = Cov(X t, X t m(t = EX t = E X e αt + σ e α(t s db s = m(e αt Cov(X s, X t = E (X s m(s(x t m(t ( s = e α(t+s E X m( + σ e αr db r (X m( + σ e αr db r s = e (V α(t+s ( + σ 2 E e αr db r e αr db r ( s t = e α(t+s V ( + σ 2 e 2αr dr = e (V α(t+s ( + σ2 2α (e2α(t s 1 V (t = (V σ2 2α + e 2αt ( σ2 2α X, σ 2 /2α Gauss, X, (σ 2 /2αe α t s, {X t } Gauss {X t } t, ξ Gauss {X t } Gauss t 1,, t n, (X(t 1,, X(t n N n Gauss (413 Φ(s R N N, B(, T -, Φ 1 (sσ(s := α(s R N n, T α(s L 2 ( B 314 {α (s} L T 1 α(s α (s T = E (α α (α α 2 ds, : = s < s 1 < s j = t, α (s := j 1 k= c ki (sk,s k+1 (s L, c k R N n, n F t -Brown B t Z (t := j 1 α (sdb s = c k (B sk+1 t B sk t Z ϕ (ζ, Brown ϕ (ζ = E e 1ζ Z j 1 = E k= k= e 1ζ c k (B sk+1 t B sk t = e 1 2 ζ ( P j 1 k= (s k+1 t s k tc k c kζ = e 1 2 ζ ( R t α (sα (s dsζ, ζ R N 1215 Z (t N Gauss Z(t := α(s db s Z (t Z(t, in L Z(t N Gauss

60 4 53 Z(t, b 1,, b n R N : = t < t 1 < < t n = t, n j=1 b j(z(t j Z(t j 1 ϕ(ζ, ϕ(ζ = E e P n 1 ζ j=1 b j (Z(t j Z(t j 1 = n E j=1 R t 1 ζb j e j t α(s db s j 1, ζ R, α { ( n } ϕ(ζ = lim exp 1 tj 2 ζ b j α (sα (s ds b j ζ j=1 t j 1 { ( n } = exp ζ 2 tj b j α(sα(s ds b j 2 t j 1 j=1, n j=1 b j(z(t j Z(t j 1 Gauss, 1217 (Z(t 1,, Z(t n Gauss {Z(t} = { } Φ 1 (sσ(s db s Gauss Gauss {Z(t} {Φ(tZ(t} Gauss, ξ Gauss, (415 Gauss 1 (411 X t = ξ + m (f i (sx s + g i (s db i s + i=1 (α(sx s + β(s ds (416 (411 f i, g i, α, β, E ξ 2 < ( (416 X t = Ψ t {ξ + ( Ψ 1 s β(s m i=1 f i (sg i (s ds + m i=1 Ψ 1 s g i (s dbs i } (417 { m Ψ t = exp f i (s db i s + i=1 ( α(s 1 2 } m f i (s ds 2 i=1 (418, Ψ t dψ t = α(tψ t dt + m f i (tψ t dbt i, Ψ = 1 (419 i=1 (418 (419 Y t := m i=1 f i (s db i s + ( α(s 1 2 m f i (s ds 2 i=1

61 4 54, {Y t } f(y = e y m ( e Yt e = e Ys f i (s dbs i + e Ys α(s 1 m f i (s ds 2 2 = m i=1 i=1 e Y s f i (s db i s e Ys d α(se Y s ds i=1 m f i (s Bs i, i=1 m f j (s Bs j e Y t = Ψ t (419 (417 X t (416 {(X t, Ψ t } f(x, y = xy 1 Ψ 1 t X t X = = + Ψ 1 s m (f i (sx s + g i (s dbs i i=1 Ψ 1 s (α(sx s + β(s ds X s Ψ 2 s X s Ψ 2 s α(sψ s ds j=1 m f i (sψ s dbs i ( m Ψ 2 s (f i (sx s + g i (sf i (sψ s ds i=1 m m Ψ 1 s g i (s dbs i + Ψ 1 s β(s ds Ψ 1 s f i (sg i (s ds i=1 i=1 i=1 2X s Ψ 3 s m f i (s 2 Ψ 2 s ds Ψ t (417 {X t } Brown, Black-Scholes i=1 43, t N(t, a(t, dn(t dt = a(tn(t, N( = N, a(t = r(t +, dn t = rn t dt + αn t db t (42 r, α, {B t } 1 Brown E N 2 <, (42 412, 425 { Ψ t = exp α db s + (r 12 } { α2 ds = exp αb t + (r 12 } α2 t

62 4 55 { N t = Ψ(t(N + + = N exp αb t + (r 12 } α2 t (421, {N t } 431 B t N, EN t = EN e rt 223 {e αbt 1 2 α2t }, (421 EN t = EN e rt Ee αbt 1 2 α2t = EN e rt Ee αb 1 2 α2 = EN e rt, {N t } 432 {N t } i r > 1 2 α2 as N t, t ii r < 1 2 α2 as N t, t iii r = 1 2 α2 as N t, lim sup N t = lim inf N t = t (421 log N t = log N + (r 12 α2 t + αb t (422 ( 214, P (Ω = 1 Ω, ω Ω ε >, t (ω t t (ω (1 + ε t B t 2t log(log t 1 + ε (422 log N + (r 12 α2 t α(1 + ε 2t log(log t log N t log N + (r 12 α2 t + α(1 + ε 2t log(log t f 1 (t = f 2 (t = (r 12 α2 t α(1 + ε 2t log(log t (r 12 α2 t + α(1 + ε 2t log(log t

63 4 56 f 1(t = (r 12 α2 α(1 + ε log(log t + 1 log t 2t log(log t r 1 2 α2, t f 2(t r 1 2 α2, t log N + f 1 (t log N t log N + f 2 (t lim f 1(t = lim f t t 2(t = r 1 2 α2, t i r > 1 2 α2 as f 1, f 2 lim log N t = t lim N t = t ii r < 1 2 α2 as f 1, f 2 lim log N t = t lim N t = t iii r = 1 2 α2 log N t = log N + αb t ( ,, Mathematica 11, pp r > 1 2 α2 ( r = 5, α = 8

64 r < 1 2 α2 ( r = 5, α = 12 3 r = 1 2 α2 ( r = 5, α = 1

65 5 ( t X t X t Z t, s t {Z s } s t, X t X t, Xt 1961 Kalman Bucy,, Riccati,, 1, 51 t X t R n X t Z t R m, dx t = b(t, X t dt + σ(t, X t du t (51 dz t = c(t, X t dt + γ(t, X t dv t (52, b : R n+1 R n, σ : R n+1 R n p, c : R n+1 R m, γ : R n+1 R m r (43 (44 {U t } X p Brown, {V t } {U t }, X r Brown, {Z s } s t X t X t 511 G t {Z s ( ; s t} σ-, σ(z s ; s t Xt {Z s } s t, Xt ( G t - (Ω, F, P (p + r Brown {(U t, V t }, (U, V =, L 2 (P := { Y ; Ω E Y 2 < } { T } L 2, T := f ;, f(t 2 dt < K = K t = K(Z, t := { Y : Ω R n ; Y L 2 (P Y G t - } (53 58

66 5 59 ( L 2 (P, L 2 T, T 2 dt 1/2 8, pp X t X t E X t X t 2 = inf Y K E X t Y 2 (54 X t 14 Hilbert (51 X t 412 Esup t T X t 2 <, T, X t L 2 (P , X t K P K (X t, P K (X t = EX t G t, 145, Xt = P K (X t, 513 X t X t G t EX t G t X t = P K (X t = EX t G t 513, Hilbert 52 1, ( dx t = F (tx t dt + C(t du t, F (t R n n, C(t R n p (55 ( dz t = G(tX t dt + D(t dv t, G(t R m n, D(t R m r (56 1, F (t, C(t, G(t, D(t R, {U t }, {V t }, X 1 Brown 521 i F, G, C, D, ii Z = iii iv X Gauss, {U t }, {V t } D(t, Xt Kalman-Bucy 522 ( 1 Kalman-Bucy 1 ( dx t = F (tx t dt + C(t du t, F (t, C(t R ( dz t = G(tX t dt + D(t dv t, G(t, D(t R

67 , X t X t = EX t G t, ( d X t = F (t G(t2 S(t X D(t 2 t dt + G(tS(t D(t 2 dz t, X = EX (57, 2 S(t = E(X t X t 2 Riccati d G(t2 S(t = 2F (ts(t dt D(t 2 S(t2 + C(t 2, S( = E(X EX 2 (58 Kalman-Bucy ( 522 Step Step (Step1 513 X t X t Z- Z sj, s j t L 2 (P L(Z, t X t P L(Z,t (X t Z-, (Step2 {Z t } 2 {N t } ( L(Z, t = L(N, t, Z-, N- (Step3 (Step4 (Step5 {N t } Brown {R t } 422 X t R t X t (Step1 Z- Z- K (53 L(Z, t L = L(Z, t := {c + c 1 Z s1 (ω + + c k Z sk (ω, s j t, c j R L 2 (P } ( X, Z s, L 2 (P n N s 1,, s n t (X, Z s1,, Z sn R n+1 Gauss P L (X = EX G = P K (X ˇX = P L (X, X = X ˇX X Gt, G G t EI G (ω(x ˇX = EI G (ω X = EI G (ωe X 1 X L(Z, t, 1 L(Z, t E X 1 = X dp = ˇX dp, ˇX G G L(Z, t ˇX G t -, ˇX = EX G t 513 X = P K (X = EX G t = ˇX = P L (X

68 5 61 X G t (1 s 1, s 2,, s n t, ( X, Z s1,, Z sn Gauss 1216, c,, c n R, c X + n i=1 c iz si Gauss ˇX L(Z, t ˇX i ˇX = a + m j=1 a jz kj, a j R, k j t ii ˇX L 2 (P - i c X + n i=1 c iz si = c (X ˇX + n i=1 c iz si X, {Z si } 1 i n, {Z kj } 1 j m Gauss 1218 Gauss L 2 (P - Gauss, ii Gauss (2 X {Z sj } 1 j n X L(Z, t E XZ sj =, E X = E X 1 = E ( X E X(Z sj EZ sj = E XZsj XEZ sj = E XZ sj EZ sj E X = (1, (2 1219, < s 1 < s 2 < < s n t X {Z s1,, Z sn } G := <s 1 < <s n t, n N σ(z s 1,, Z sn, σ( X G σ( X σ(g, Dynkin 112 D := {B ; P (B A = P (BP (A, A σ( X}, G D, D Dynkin A σ( X, i P (Ω A = P (A = P (ΩP (A Ω D ii B 1, B 2 D, B 1 B 2 P ((B 2 \ B 1 A = P (B 2 A P (B 1 A = P (B 2 P (A P (B 1 P (A = P (B 2 \ B 1 P (A iii B 2 \ B 1 D B n D, n = 1, 2, B n B, P (B A = lim n P (B n A = lim n P (B n P (A = P (BP (A B D

69 5 62 G intersection, Dynkin σ(g D σ( X σ(g = G t 523, 524 X t (55, ( X t M t := R 2 {M t } Gauss Z t M t 2 Markov ( X dm t = H(tM t dt + K(tdB t, M =, 2 Brown {B t } H(t, K(t ( ( ( U t F (t C(t B t =, H(t =, K(t = V t G(t D(t 521 M Gauss, 424 {M t } Gauss 524, 1216 (X t, Z s1,, Z sn Gauss, 523 X t = P K (X t = EX t G t = P L (X t (51 X t Z-, Z- (Step2 Step, {Z t } {N t }, L(Z, t = L(N, t ( 527, {N t } {Z t } 525 ( N t := Z t P L(Z,s (G(sX s ds = Z t G(s X s ds (511 {N t } {Z t } 526 {N t } 2 G t - Z t Gt EZ t G t = E G(sX s dsg t + E D(s dv s (512

70 5 63 (1, (2 (1 EZ t G t = Z t (2 E G(sX s dsg t = G(sE X s G t ds = G(s X s ds (1 {Z s } s,t G t -, 143 (2 G(sE X s G t B(, t G t - G(s B(, t - ω EX s G t i EX s G t t G t - ii X s (ω as, s n s X sn X s as, 412 X sn sup n X sn L 1 (P, Lebesgue 143 EX sn G t EX s G t as EX s G t t s i, ii 211, EX s G t B(, t G t -, G(s, t K, B G t E G(sEX s G t ; B K E EX s G t K E E X s Gt = K E X s K E sup X s s t < E G(sEX s G t ; B ds tk E sup X s s t <, 129, ( E G(sE X s G t ds ; B = G(sE X s G t ds dp B ( ( = G(sE X s G t dp ds = G(s E X s G t dp ds B B ( ( = G(s X s dp ds = G(sX s ds dp = E G(sX s ds ; B B B, (2 (1,(2 (512 N t = Z t G(s X Gt s ds = E D(s dv s, H s,t := σ((u u, V u ; s u t, V t,t := σ(v β V α ; t α β T (513

71 5 64 G t H,t, H,t V t,t Brown D(sdV s V t,t - G t T t, T T Gt E D(s dv s = E D(s dv s = t T Gt N t = E D(s dv s t M T := T D(s dv s D(s, T, 2 222, N t = EM T G t {N t } G t - E N t 2 E E M T 2 Gt = E MT 2 < L(N, t = L(Z, t, 528 f L 2, T A 1, A 2 >, T ( T 2 T A 1 f(t 2 dt E f(t dz t A 2 f(t 2 dt f L 2, T ( T 2 ( T T 2 E f(t dz t = E f(tg(tx t dt + f(td(t dv t ( T 2 ( T 2 = E f(tg(tx t dt + E f(td(t dv t := I 1 + I 2 + I 3 ( T +2E Schwarz T T I 1 f(t 2 dt E ( T f(tg(tx t dt f(td(t dv t G(tX t 2 dt G(t, T, Fubini 412, K T T E G(tX t 2 dt K 2 E X t 2 dt T = K 2 E X t 2 dt K 2 T E sup X t 2 < t T

72 5 65 A 2 I 1 A 2 T f(t 2 dt, 318, D(t, T, A 2 T T I 2 = E f(td(t dv t = E f(t 2 D(t 2 dt = T f(t 2 D(t 2 dt A 2 T f(t 2 dt {X t } {V t }, T I 3 = E, A 2 = A 2 + A 2 ( T 2 E f(t dz t T f(tg(tx t dt E f(td(t dv t = T A 2 f(t 2 dt, ( T 2 E f(t dz t = ( T 2 E f(td(t dv t T f(t 2 D(t 2 dt T inf t T D(t2 f(t 2 dt, 521, T < inf t T D(t2 <, A 1 > T ( T 2 A 1 f(t 2 dt E f(t dz t, L(Z, T, Z t { 529 N (Z, T := c + } T f(t dz t ; f L 2, T, c R L(Z, T = N (Z, T i N (Z, T L(Z, T ii iii N (Z, T c + k i=1 c iz ti, t i T N (Z, T L 2 (P

73 5 66 ii, iii L(Z, T N (Z, T, i L(Z, T = N (Z, T i f L 2, T, f {f n } n N, f n (t = k n i= c ii (t n i,t n i+1 (t f n (t f(t in L 2, T 528 T T 2 T 2 E f(t dz t f n (t dz t = E (f(t f n (t dz t T A 2 f(t f n (t 2 dt, n L 2 (P - lim 2, T f(t dz t = L 2 (P - lim n k n i= N (Z, T L(Z, T c i (Z t n i+1 Z t n i L(Z, T ii = t t 1 < < t k = T Z j := Z tj+1 Z tj, c j k c i Z ti = i=1 = k 1 k 1 c j Z j = j= ( T k 1 j= j+1 j= t j c ji (tj,t j+1 (t c j dz t dz t N (Z, T ii T iii f n(t dz t M in L 2 (P L 2 (P M L 2 (P, { T f } n(t dz t L 2 (P Cauchy 528, m, n T T 2 A 1 f m (t f n (t 2 dt E (f m (t f n (t dz t = E T f m (t dz t T f n (t dz t 2 (, {f n } L 2 T, T 2 dt 1/2 Cauchy L 2, T, f(t L 2 T, T lim f n n f 2 dt = 528, n E T f n (t dz t T 2 T f(t dz t A 2 f n (t f(t 2 dt T f n(t dz t T f(t dz t in L 2 (P M = T f(t dz t as M N (Z, T iii

74 (r, s- (B(R B(R- g(r, s i g(r, s Volterra, s > r g(r, s = ii iii T T g(r, s 2 dr ds < G(r X r = c (r + r g(r, s dz s, i, ii g(r, s Volterra, Volterra f(s = h(s + Volterra s g(r, sf(r dr, h(s L 2, t G(r X r = P L(Z,r (G(rX r L(Z, r 529, g(s L 2, r, c (r R G(r X r = c (r + r g(s dz s, g(s { g(s (s r g(r, s := (r < s g(r, L 2, T g(r, s Volterra g(r, s 2 (r, s u r, T G(r X r = c (r+ u g(r, s dz r s, u, r g(r, s dz u s V u, u, T ( r ( u EG(r X r V u = E c (r + g(r, s dz s V u = E c (r + g(r, s dz s V u u u = Ec (rv u + E g(r, sg(sx s ds V u + E g(r, sd(s dv s V u,, X s V u 322 EG(r X r V u u u = E g(r, sd(s dv s dv s u u = E g(r, sd(s dv s, dv s u u = E g(r, sd(s ds = g(r, sd(s ds (514, Xr (r, ω- (B(R F- X r, as X r = X r ( 9, p8 X r (r, ω- Brown, V u (u, ω- u

75 5 68 ((r, ω, u X r (ωv u (ω u (r, ω X r (ωv u (ω (r, ω-, (r, ω u X r (ωv u (ω 123 X r (ωv u (ω (r, u, ω- Fubini E X r V u (r, u- (514 (r, u-, u g(r, sd(sds (r, u- 1 s+ 1 n D(sḡ(r, s := lim sup g(r, ud(u du n n s ( s+ 1 = lim sup n 1 n n g(r, ud(u du s g(r, ud(u du (r, s- D(sḡ(r, (r, s-, s D(sḡ(r, s r s g(r, ud(u du Radon-Nikodym, D(sḡ(r, s = D(sg(r, s as s- D(s > ḡ(r, s = g(r, s as, g(r, s (r, s- ḡ(r, s g(r, s (r, s-, i, iii ii E G(r X r 2 r inf s T D(s2 g(r, s 2 ds (515 E r g(r, sd(s dv s =, Xs V s, r E G(r X r g(r, sd(s dv s ( r r r = E c (r + g(r, sg(sx s ds + g(r, sd(s dv s g(r, sd(s dv s r r = E g(r, sd(s dv s = g(r, s 2 D(s 2 ds, Schwarz, r E G(r X r g(r, sd(s dv s E = E G(r X r E G(r X 1 ( r r 2 2 ( r ( r g(r, s2 D(s 2 ds 1 2 E G(r X 1 ( r 1 r 2 2 g(r, s 2 D(s 2 2 ds E G(r X r 2 r g(r, s 2 D(s 2 ds g(r, sd(s dv s g(r, s 2 D(s 2 ds r inf s T D(s2 g(r, s 2 ds 1 2

76 5 69 (515, (515 T T g(r, s 2 ds dr = T r ( C g(r, s 2 ds dr 1 T inf E E X r 2 dr CT E s T D(s2 T G(r X T r 2 dr C sup X r 2 < r T E X r 2 dr ii 527 i L(N, t L(Z, t L(N, t c +, L 2 (P - c + n c k N tk = c + k=1 n c k N tk (516 k=1 n c k Z tk k=1 n k c k G(s X s ds (517, G(s X s = P L(Z,s (G(sX s L(Z, s L(Z, t k G(s X s ds = L 2 (P - lim i k=1 G(s i X si (s i+1 s i L(Z, t, (516 L(Z, t (516 L 2 (P - (517 L 2 (P -, L(Z, t L(Z, t L(N, t L(Z, t ii L(N, t L(Z, t 521, f L 2, T f(s dn s = = f(s dz s f(s dz s f(rg(r X r dr ( r f(r g(r, s dz s dr f(rc (r dr ( r ( f(r g(r, s dz s dr = f(rg(r, s dz s dr ( ( = f(rg(r, sg(sx s ds dr + f(rg(r, sd(s dv s dr

77 5 7 (1 G(s, t, X s (ω ω s,, t K 1 := K 1 (ω G(sX s K 1 Schwarz f(rg(r, sg(sx s dr ds K t 1 f(rg(r, s dr ds ( 1 ( K 1 f(r 2 2 t 1 dr g(r, s 2 2 dr ds, f L 2, T 521, 129 Fubini, (2 D(s K 2 Schwarz 2 ( ( E f(rg(r, sd(sdr d V s K 2 2 E f(r 2 dr g(r, s 2 dr ds <, Lebesgue-Stieltjes 5212 (, ( f(s dn s = f(s f(rg(r, s dr dz s f(rc (r dr (518 Volterra, h L 2, t Volterra f(s = h(s + s g(r, sf(r dr f(s L 2, t ( 5, pp46 51, h(s = I,ti (s, t i t, f i I,ti (s = f i (s, (518 Z ti s f i (s dn s = Z ti = g(r, sf i (r dr = f i (s I,ti (s dz s f i (rc (r dr + g(r, sf i (r dr f i (rc (r dr f i (s dn s, : = s < < s n = t, f i L 2, t f i (rc (r dr < sup c (r f i (r dr < r t n f i (s dn s = L 2 (P - lim f i (s j (N sj+1 N sj L(N, t j=

78 5 71 Z ti L(N, t, L(Z, t L(N, t i, ii L(Z, t = L(N, t , p84, {N t } {Z t }, {N t } σ(z s ; s t = σ(n s ; s t Gauss, 527 {N t } {Z t } 5212 (, T Ω R 1 ((s, ω, a ϕ(s, a, ω M M 2,c t, T, { ϕ(s, a, ω dm s} M2,c, (a, ω ϕ(s, a, ω dm s B(R 1 F t - 2 E ϕ(s, a, da d M R s < 1 { } ϕ(s, a, ω da R 1 dm s = 3, pp R 1 { } ϕ(s, a, ω dm s da (Step3 Brown D(t, {R t (ω} R t (ω = 5213 {R t } 1 G t -Brown 1 D(s dn s(ω (519 1, t, {R D(s t} 2, N t, R t = dn t = dz t G(t X t dt = G(t(X t X t dt + D(t dv t (52 1 D(s dn s = 1 t D(s 2 d N 1 s = D(s 2 D(s2 ds = t (521 Lévy 336, {R t } 1 G t -Brown L(R, t = L(N, t L(R, t = L(N, t = L(Z, t X t = P L(R,t (X t (522 X t R t

79 X t, Radon-Nikodym X t = EX t + s s EX tr s dr s (523 R t G =, D = 1, V = R Z t 529, g L 2, t, c (t R, X t = c (t + g(s dr s (524 (524 E X t = c (t, E X t = E EX t G t = EX t c (t = EX t, f L 2, t (X t X t f(s dr s L(R, t, E X t f(s dr s = E X t f(s dr s = E g(s dr s f(s dr s + E c (t f(s dr s = E g(s dr s, f(s dr s = g(sf(s ds f(s = I,r (s, r t EX t R r = E X t I,r (s dr s = r g(s ds, g( EX t R Radon-Nikodym g(r = r EX tr r, (Step4 X t dx t = F (tx t dt + C(tdU t 422 X t = e R t F (s ds (X + = e R t F (s ds X + e R s F (u du C(s du s e R t s F (u du C(s du s (525 EX t = EX e R t F (s ds (526