[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo

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Download "[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo"

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1 @phykm [2] [2] [1] ([4] ) 1 Ω = 2 N {Π n =1 A { 1, 1} N n N, A {{ 1, 1}, { 1}, {1}, }} B : Ω { 1, 1} P (Π n =1 A 2 N ) = 2 #{ A={ 1},{1}} X = j=1 B j B X +k X V[X ] = 1 ( ) 1 1 dt dx W (t) = t/dt =1 dxb (1) B 1 W (t) dx 2 (t/dt) σ dx 2 (t/dt) σ 2 t dx = σ dt W (t) W (t) W (t + s) W (t) [t, t + s] W (t + s) W (t) s W () = W (t + s) W (s) σ 2 s 1

2 [1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Proposton 1.4. Proof. 2 E[W (s)w (t)] = σ 2 mn{s, t} E[(W (t) W (s)) 4 ] = 3(t s) 2 σ 4 Kolmogorov [3] Ω R concrete Ω ω Ω [2] 1,2 t s E[W (t)w (s)] (2) =E[(W (t) W (s))w (s) + W (s) 2 ] (3) =E[W (t) W (s)]e[w (s)] + σ 2 s (4) =σ 2 s (5) *1 2

3 E[(W (t) W (s)) 4 ] (6) =E[W (t s) 4 ] (7) 1 = x 4 exp( x 2 /2(t s)σ 2 )dx (8) 2σ2 π(t s) (t s)σ2 = 2σ2 π(t s) ([x3 exp( x 2 /2(t s)σ 2 )] 3x 2 exp( x 2 /2(t s)σ 2 )dx) (9) (t s)σ2 = 3x 2 exp( x 2 /2(t s)σ 2 )dx (1) 2σ2 π(t s) = (t s)2 σ 4 2σ2 π(t s) ([3x exp( x2 /2(t s)σ 2 )] 3 exp( x 2 /2(t s)σ 2 )dx) (11) =3(t s) 2 σ 4 (12) dtσ 2 V[(W (t) W (s)) 2 σ 2 t s ] (13) =E[((W (t) W (s)) 2 σ 2 t s ) 2 ] (14) =E[(W (t) W (s)) 4 ] 2E[(W (t) W (s)) 2 ] t s σ 2 + σ 4 t s 2 (15) =(3 t s 2 2 t s 2 + t s 2 )σ 4 (16) =2 t s 2 σ 4 (17) Proposton 1.5. W (t) 1. X 1 (t) = hw (t/h 2 ) 2. X 2 (t) = W (t + h) W (h) 3. X 3 (t) = tw (1/t), X 3 () = 4. X 4 (t) = W (t) Proof. (1): (2): (3): W ( ) E[X 3 (t)x 3 (s)] = tsmn{1/t, 1/s} = mn{t, s} (18) (4): W (t) Defnton U(t) = exp( µt)w (σ 2 exp(2µt)/2µ) 3

4 Defnton 1.7. X(t) = exp(σw (t)) p X (y) = y log yp W (log y) (19) 1 = 2πσ2 ty exp (log y)2 2σ 2 t (2) Defnton 1.8. W (1) = 1 [, 1] Cov[X(s), X(t)] = s(1 t), (s t) W (t) W (t) tw (1), (1 t)w (t/(1 t)) * f, g fdg = lm sup t +1 t f(s )(g(t +1 ) g(t )), (s [t, t +1 )) (21) f, g s dw (t) 2 dt o(dt) dw (t) 2 o(dt) (Ω, F t F, P ) F t {W (s) s t} (Ω, F) Defnton 2.1. V f : Ω R + R (Ω, F) (R +, B) H t E[W (t) H s ] = W (s), (s t) f(, t) H t ( H ) t :E[ f 2 dt] < *2 4

5 H t E[W (t) H s ] = W (s) H s F t F t H t F t [2] [1] L 2 Defnton 2.2. f V f(ω, t) = n X (ω)1 [t,t +1)(t) V Defnton 2.3. f V f(ω, t) = n X (ω)1 [t,t +1)(t) f(, t)dw (t) = X (W (t +1 ) W (t )) (22) L 2 (Ω), L 2 (Ω, R + ) 1. dw (t) : V L 2 (Ω) (V L 2 (Ω, R + ) ) E[( f(, t)dw (t)) 2 ] = E[ f 2 (, t)dt] (23) 2. E[ f(, t)dw (t)] = 3. f(, t)dw (t) H t (f ) 4. E[ f(, t)g(, t)dw (t)] = E[ f(, t)dw (t) 3 2 H t g(, t)dw (t)] (24) E[ X (W (t +1 ) W (t ))] (25) = = = E[E[X (W (t +1 ) W (t )) H t ]] (26) E[X E[(W (t +1 ) W (t )) H t ]] (27) E[X ] (28) = (29) 1 L 2 X H t E[( X (W (t +1 ) W (t ))) 2 ] (3) = j E[X X j (W (t +1 ) W (t ))(W (t j+1 ) W (t j ))] + E[X 2 (W (t +1 ) W (t )) 2 ] (31) 5

6 E[X 2 (W (t +1 ) W (t )) 2 ] (32) =E[E[X 2 (W (t +1 ) W (t )) 2 F t ]] (33) =E[(W (t +1 ) W (t )) 2 E[X 2 F t ]] (34) =E[(W (t +1 ) W (t )) 2 ]E[E[X 2 F t ]] (35) =E[(t +1 t )X 2 ] (36) f k = max, j H k X(t k+1 ) X(t k ) 4 f, g V V L 2 (Ω, R + ) L 2 (Ω, R + ) Proposton 2.4. f V L 2 (Ω, R + ) f n Proof. V V V V V V ω Ωpontwse V V V : ω Ω f V f n (ω, t) = f(ω, t )1 [t,t +1) dt f n f V V : E[ (f f n ) 2 dt] (37) [1] suppϕ n [ 1/n, ], ϕ n = 1, ϕ n ω Ωpontwse L 2 ϕ n f f L 2 V V: [ n, n] Proposton 2.5. T o W (t)dw (t) = 1 2 W (T )2 1 2 T (38) 6

7 Proof. N(t) = W (t ), (t [t, t +1 )) L 2 W (t) E[ =E[ tn (N(t) W (t)) 2 dt] (39) t+1 t (W (t) W (t )) 2 dt] (4) = = = t+1 t E[(W (t) W (t )) 2 ]dt (41) t+1 (t t )dt (42) t (t +1 t ) 2 (43) N(t) = N(t)dW (t) (44) W (t )(W (t +1 ) W (t )) (45) = 1 2 (W (t +1) + W (t ))(W (t +1 ) W (t )) 1 2 (W (t +1) W (t ))(W (t +1 ) W (t )) (46) = 1 2 W (t n) (W (t +1) W (t )) 2 (47) 1 2 W (t n) t n E[( 1 ((W (t +1 ) W (t )) 2 (t +1 t ))) 2 ] (48) 2 E[( 1 ((W (t +1 ) W (t )) 2 (t +1 t ))) 2 ] (49) 2 E[((W (t +1 ) W (t )) 2 (t +1 t )) 2 ] (5) (t +1 t ) 2 (51) t T W (t)dw (t) = 1 2 W (T )2 1 2 T (52) 7

8 Proposton 2.6. f(t) t T T f(t)2 dt f(t)dw (t) (53) Proof. f(t) L 2 (R + ) f = f 1 [t,t +1) f(t) L 2 (R + ) f dw (t) = f (W (t +1 ) W (t )) (54) f 2 (t +1 t ) *3 f 2 (t +1 t ) f 2 dt Proposton 2.7. f( Ω ) T T f(t)dw (t) = f(t )W (T ) W (t)df (55) Proof. ( ) f(t )W (T ) = = n f(t +1 )W (t +1 ) f(t )W (t ) (56) n f(t )(W (t +1 ) W (t )) + (f(t +1 ) f(t ))W (t +1 ) (57) f L 2 n f(t )(W (t +1 ) W (t )) = T f(t)dw (t) (58) f f(t + dt) f(t) = dtf (t + dt) + o(dt) (f(t +1 ) f(t ))W (t +1 ) = f (t +1 )W (t +1 )(t +1 t ) + o(t +1 t ) (59) ω Ω * 4 T W (t)df = T f W (t)dt (6) = σ 2 *3 *4 8

9 3 dx(t) = Y (t)dt + Z(t)dW (t) (61) t d( ) X() X(t) = X() + t Y (s)ds + t Z(s)dW (s) (62) 3 X, Y, Z, W Brown dw dt 2 Defnton 3.1. W (t) W (t) µ(t), σ(t) ω Ω µ(ω)( ) L 1 (R + ), σ(ω)( ) L 2 (R + ) X(t) = X() + t µ(s)ds + t σ(s)dw (s) (63) dx(t) = µ(t)dt + σ(t)dw (t) (64) Proposton 3.2. Proof. X(t + dt) X(t) = t+dt µ(s)ds + t t t+dt σ(s)dw (s) (65) ωpontwse µ(s) L 1 t+dt t+dt µ(s)ds µ(s) dt (66) E[ t t+dt t 2 σdw (s) ] = E[ t t+dt t σ(s) 2 ds] (67) 9

10 Theorem 3.3. F (, ) : R R + R C 2 dx(t) = µ(t)dt + σ(t)dw (t) (68) X(t) = X() + t µ(s)ds + F (X(t), t) t σ(s)dw (s) (69) df (X(t), t) = F t (X(t), t)dt F 2 x 2 (X(t), t)σ(t)2 dt + F (X(t), t)dx(t) (7) x F x (X(t), t)dx(t) F F (X(t), t)µ(t)dt + (X(t), t)σ(t)dw (t) (71) x x dx X(t) F (X(t), t) F (X(t), t) t dt dw dw dt dt 2, dtdw o(dt) dw 2 dt F Proof. F (X(t), t) µ, σ ωpontwsen µ, σ L 1, L 2 µ, σ L 1, L 2 t µ(ω, t) = X (ω)1 [t,t +1)(t)σ(ω, t) = Y (ω)1 [t,t +1)(t) ωpontwse (1,2 ) ω Ω X(t) dt dx = X(t + dt) X(t) dt, dx = µdt + σdw F (X + dx, t + dt) F (X, t) (72) =( x F )µdt + ( x F )σdw + ( t F )dt (73) + 1 [ ( t 2 2 )dt 2 + ( x 2 F )µ 2 dt 2 + ( x t F )µdt 2 (74) + ( 2 x F )σ 2 dw 2] (75) +( 2 x F )µσdtdw + t x F σdtdw (76) F F (X, t) F F (X + θdx, t + θ dt), (θ, θ [, 1]) df = ( t F )dt ( 2 xf )σ 2 dt + ( x F )µdt + ( x F )σdw (77) 1

11 F (X(t), t) F () = t ( ( t F ) + 1 ) 2 ( 2 xf )σ 2 + ( x F )µ ds + t ( x F )σdw (78) [, T ] (72) (73) ωpontwse ( x F )µdt + ( t F )dt t (( t F ) + ( x F )µ) ds (79) ( x F ) ( x F )σdw t ( x F )σdw (8) L 2 (74) dt 2 dt = t t 1 n (t t 1 ) 2 (81) =1 (74) (75) L 2 ( x 2 F )σ 2 dw 2 t ( 2 xf )σ 2 ds (82) ( 2 x F )σ 2 dw 2 t ( 2 xf )σ 2 ds L 2 (83) ( x 2 F )σ 2 dw 2 ( 2 F x )σ 2 dt L 2 (84) + ( 2 x F )σ 2 dt ( 2 xf )σ 2 dt L 2 (85) + ( 2 xf )σ 2 dt t ( 2 xf )σ 2 ds L 2 (86) (85) [, t] 2 x F, 2 xf (86) 11

12 (84) ( 2 x F )σ 2 (dw 2 dt ) 2 L 2 (87) =,j E[( 2 x F )σ 2 ( 2 x F j )σ 2 j (dw 2 dt )(dw 2 j dt j )] (88) = E[( 2 x F ) 2 σ 4 (dw 2 dt ) 2 ] (89) = E[( 2 x F ) 2 σ 4 ]E[(dW 2 dt ) 2 ] (9) = E[( 2 x F ) 2 σ 4 ]2dt 2 (91) E[( 2 x F ) 2 σ 4 ] dt2 4 dtdt = (92) dw dw = dt (93) dw dt = (94) d( ) dtdt = (95) dm dm j = d M, M j (96) dmdt = (97) M, M j L 2 t *5 [2] / *5 1 12

13 [1] B. (212) [2] - - (214) [3] (25) [4] (24) 13

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