21世紀の統計科学 <Vol. III>

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1 21 III HP, ( ),.,,.,,. 1 uchida@sigmath.es.osaa-u.ac.jp 168

2 1, (stochastic differential equation, SDE) 1 X. dx t = b(x t,α)dt + σ(x t,β)dw t, t [,T], X = x., b : R Θ α R, σ : R Θ β R w 1 Wiener, x. b, σ. α β ( ), b σ., Θ α Θ β, R., X T := {X t } t [,T ] X n := {X hn } =,1,...,n, nh n = T,.,,.,.,,. Yoshida [31] h n, nh n nh 3 n,, α β,., α β. Kessler [14] h n, nh n nh l n (l 2), Ito-Taylor, α β,. [,T], h n = T/n, α,.,,., SDE 1. dx t = b(x t,α)dt + εσ(x t,β)dw t, t [,T], ε (, 1], X = x, ε., Genon-Catalot [4] Laredo [21] β, α 169

..,n, nh n = T,.,,.,.,,. Yoshida [31] h n, nh n nh 3 n,, α β,., α β. Kessler [14] h n, nh n nh l n (l 2), Ito-Taylor, α β,.

3 . Sørensen and Uchida [28] α β,.,,., 1, 1,. 2,,. 3,. 4,., 5.., 2, 3, (Ω, F,P; {F t } t ), 1 F t - Wiener {w t } t., , 1 X. dx t = b(x t,α)dt + σ(x t,β)dw t, t [,T], X = x, (2.1), b : R Θ α R, σ : R Θ β R, w 1 Wiener, x., θ := (α, β) Θ:=Θ α Θ β R 2, Θ. θ =(α,β ) SDE (2.1), θ Int(Θ). 1. (Ornstein-Uhlenbec ). b(x, α) = αx, σ(x, β) =β, dx t = αx t dt + βdw t, t [,T], X = x, (2.2) α>, β>. 17

4 2. (The hyperbolic diffusion process). b(x, α) = α x 1+x 2, σ(x, β) =β, X t dx t = α dt + βdw t, t [,T], X = x, (2.3) 1+X 2 t α>, β>. 2.2 SDE (2.1) θ = θ dx t = b(x t,α )dt + σ(x t,β )dw t, t [,T], X = x (2.4) t n = h n,( =, 1,...,n)., t n n = nh n = T, h n., X n = {X t n } =,1,...,n. 3. (i) (nh n = T h n ). (ii) (h n =Δ nh n = nδ =T ). (iii), (h n nh n = T ). (i),. (ii),. (iii) ( (i)) ( (ii)),. 2, (iii)., n, h n T = nh n. 3. (1985/1/1 24/12/31 2 ). T = 2, h n = T/n = 2/(1461 5) = 2/ nh 2 n = Th n 2.27 = (2/1/1 24/12/31 6 ). T =6, h n = T/n =6/ (i) nh 2 n = Th n 6.32 = (ii) nh 3 n = Th 2 n (1985/1/1 24/12/31 2 ). T = 2, h n = T/n =2/ (i) nh 2 n = Th n 2.83 = (ii) nh 3 n = Th 2 n (iii) nh 4 n = Th 3 n

(iii), (h n nh n = T ). (i),. (ii),. (iii) ( (i)) ( (ii)),. 2, (iii)., n, h n T = nh n. 3. (1985/1/1 24/12/31 2 ). T = 2, h n = T/n = 2/(1461 5) = 2/735.27. nh 2 n = Th n 2.27 =.54. 4.

5 (T h n ), T. 4, T =5( ).,, h n. 3 : Th n = nh 2 n. 4 : Th 2 n = nh 3 n. 5 : Th 3 n = nh 4 n. 2.3 (simulation) SDE (2.1) X = {X t } t [,T ] ( )., T., α,β. t>s, X t X s = t s b(x u,α )du + t s σ(x u,β )dw u, X = x. X t, X (n) t. X (n) t n X (n) t n X (n) t n = b(x (n) t n,α )(t n t n )+σ(x (n) t n,β )(w t n w t n ), = x.., h n = t n tn, ΔW t n := w t n w t n N(,h n ), =1,...,n, X (n) t n = x t n + b(x t n,α )h n + σ(x t n,β )ΔW t n, X (n) t n = x t, X X (n) = {X (n) t n,x(n) t n,...,x(n) 1 t }., ΔW n t n, ΔW t n,...δw 1 t n. X (n) T = X (n) t n, [ ] (Kloeden and Platen [15]) c> XT, E X (n) c h n. T 6. (Ornstein-Uhlenbec ) dx t = αx t dt + βdw t, X = x, t [,T]. 7-1 ( ) x =6, α =1, β =2, T =2, h n =1/1., 1/2. 172

., h n = t n tn, ΔW t n := w t n w t n N(,h n ), =1,...,n, X (n) t n = x t n + b(x t n,α )h n + σ(x t n,β )ΔW t n, X (n) t n = x t, X X (n) = {X (n) t n,x(n) t n,...,x(n) 1 t }.

6 6 7-1: , X = {X t } t [,T ], ,,, ( ). 2.4 SDE (2.1) X (state space) (, ). X θ = θ, μ θ, f(x) μ θ (dx) < f, T, 1 T f(x t )dt f(x)μ θ (dx) 1., μ θ θ = θ X,, t> 1 B, μ θ (B) = P x(x t B)μ θ (dx)., P x x X.. 173

7 X scale measure s(x, θ)dx speed measure ξ(x, θ)dx. x R, θ Θ, { x } 2b(y, α) s(x, θ) := exp σ 2 (y, β) dy, { 1 x } ξ(x, θ) := σ 2 (x, β) exp 2b(y, α) σ 2 (y, β) dy 1 = σ 2 (x, β)s(x, θ). x = / x, δ α = / α, δ β = / β. R Θ f(x, θ), δ θ f(x, θ) =(δ α f(x, θ),δ β f(x, θ)).. C,l (R Θ α) f : (i)f(x, α) R Θ α, x, n =, 1,...,, C>, x, sup α x n f(x, α) C(1 + x ) C. (ii) n =, 1,...,, x n f(x, α) α l, ν =, 1,...,l, C>, x sup α δα ν x n f(x, α) C(1 + x ) C. 2 2 (Fisher ) I(θ ) ( ) I b (θ ) I(θ )= I σ (θ )., I b (θ ) = I σ (θ ) = 2 ( ) 2 δα b(x, α ) μ θ(dx), σ(x, β ) ( ) 2 δβ σ(x, β ) μ θ(dx). σ(x, β ) P θ SDE(2.1) X, E θ P θ. p d P θ.. A1 θ Θ, s(x, θ)dx =, s(x, θ)dx =, 174

R Θ f(x, θ), δ θ f(x, θ) =(δ α f(x, θ),δ β f(x, θ)).. C,l (R Θ α) f : (i)f(x, α) R Θ α, x, n =, 1,...,, C>, x, sup α x n f(x, α) C(1 + x ) C. (ii) n =, 1,.

8 Ξ(θ) = ξ(x, θ)dx <. A2 (i) L>, x, y R, θ Θ b(x, α) b(y, α) + σ(x, β) σ(y, β) L x y. (ii) b(x, α) C 2,3 (R Θ α). (iii) σ(x, β) C 2,3 (R Θ β),, inf x,β σ 2 (x, β) >. (v) p, sup t E θ [ X t p ] <. A3 (i) I(θ ). (ii) x, b(x, α) =b(x, α )= α = α. x, σ 2 (x, β) =σ 2 (x, β )= β = β., (, Kutoyants [19]). 1. A1, A2-(i), x, β, σ 2 (x, β) >., X., μ θ (x) ξ(x, θ)/ξ(θ). A2-(i) SDE(2.1) (5 6). A3-(ii) (identifiability condition),. 7. (Ornstein-Uhlenbec ) scale measure s(x, θ)dx speed measure ξ(x, θ)dx { } α s(x, θ) = exp β 2 x2, ξ(x, θ) = 1 { β exp α } 2 β 2 x2, α, β > { α exp Ξ(θ) = β 2 x2 } dx =, 1 β 2 exp { α exp { αβ 2 x2 } dx = β 2 x2 } dx =, π αβ 2 <., 1 Ornstein-Uhlenbec ( 1), μ θ { } ξ(x, θ) μ θ (dx) = Ξ(θ) dx = 1 πβ2 /α exp x2 dx β 2 /α,, β 2 /(2α). 175

, μ θ (x) ξ(x, θ)/ξ(θ). A2-(i) SDE(2.1) (5 6). A3-(ii) (identifiability condition),. 7.

9 8. (Ornstein-Uhlenbec ) SDE(2.2), t ) X t = e (X αt + βe αs dw s. (2.5), X t, (5 5),. f(t, v) =e αt v, V t = X + t βeαs dw s,, X t = f(t, V t )=V α = X α t X s ds + t t e αs V s ds + βdw s. t e αs βe αs dw s, X t E θ [X t ] V θ [X t ] (2.5) 5 4, E θ [X t ] = x e αt, V θ [X t ] = β 2 1 e 2αt. 2α X t, p sup t E θ [ X t p ] <. 9. (Ornstein-Uhlenbec Fisher ) 7, μ θ N(,β 2 /(2α)). I(θ ) = ( 1 β 2 x2 μ θ (dx) 2 β 2 α,β >, det(i(θ )) = 1 α β 2. ) = ( 1 2α 2 β 2 ). 2.5 (2.1), X t = x X t y p(h n,x,y; θ). X n =(X t n,x t n 1,...,X t n ) (X t n = x ) f Xt n,x t n 1,...,X t n n (x t,x t1,...,x tn ; θ) = n p(h n,x t,x t ; θ) =1 176

5) 5 4, E θ [X t ] = x e αt, V θ [X t ] = β 2 1 e 2αt. 2α X t, p sup t E θ [ X t p ] <. 9. (Ornstein-Uhlenbec Fisher ) 7, μ θ N(,β 2 /(2α)).

10 , l n (θ) = n log p(h n,x t n,x t n ; θ) =1. ˆθ (ML) n l n (ˆθ (ML) n )=supl n (θ) θ. p,,. l n (θ) θ, S n (θ) :=δ θ l n (θ) = n δ θ p(h n,x t n,x t n ; θ) =1 p(h n,x t n,x t n ; θ) R ˆθ (ML) n, S n (ˆθ n (ML) )=. F n {X t n } =,1,...,n σ-,, F n = σ(x t n,x t n 1,...,X t n ),, [ ] δθ p(h n,x t n E θ [S n (θ) S (θ) F ] = E,X t n ; θ) θ p(h n,x t n,x t n ; θ) X t n δ θ p(h n,x t n =,y; θ) p(h n,x t n,y; θ) p(h n,x t n,y; θ)dy = R = δ θ δ θ p(h n,x t n,y; θ)dy R p(h n,x t n,y; θ)dy =., S n (θ) F n -.,,, h n nh n, (ML) ˆθ n., m(x, θ) :=E θ [X t n X t n = x], v(x, θ) :=V θ [X t n X t n = x]. 1. (Ornstein-Uhlenbec ) 8, t X t = x e αt + βe αt e αs dw s. 177

11 t eαs dw s Wiener,, t e2αs ds = 1 2α (e2αt 1)., { 1 p(t, x, y; θ) = πβ2 (1 e 2αt )/α exp (y } e αt x) 2. β 2 (1 e 2αt )/α, X = x X t L(X t X = x) ( ) L(X t X = x) N e αt x, β2 (1 e 2αt ) 2α, l n (θ) = 1 2 n =1 { log(2π) + log + (X t n } e αhn X t n ) 2 β 2 (1 e 2αh n )/(2α) ( β 2 (1 e 2αh n ) ) 2α (2.6)., m(x, θ) v(x, θ), L(X t n m(x, θ) = e αh n x, v(x, θ) = β2 (1 e 2αhn ) 2α X t n = x) N (m(x, θ),v(x, θ)), (2.6),. { l n (θ) = 1 n log(2π) + log v(x t n 2,θ)+ (X t n m(x } t n,θ))2. v(x t n,θ) =1 1, Ornstein-Uhlenbec p (ML), ˆθ n., Ornstein- (ML) Uhlenbec, h n nh n, ˆθ n., SDE (2.1) X., SDE 178

2 (1 e 2αh n )/(2α) ( β 2 (1 e 2αh n ) ) 2α (2.6).

12 ,., 1, L(X t n X t n = x) m(x, θ), v(x, θ). U n (θ) = n U(X t n,x t n,θ), (2.7) =1 U(x, y, θ) = 1 2 (y m(x, θ))2 log(2πv(x, θ)). (2.8) 2v(x, θ) U n (θ),., (M) U n (θ). ˆθ n U n (ˆθ n (M) )=supu n (θ)., U n (θ) θ, θ ( ) ( ) M n (α) (θ) δ α U n (θ) M n (θ) = := = (θ) δ β U n (θ) M (β) n M-.,, M n (θ) = n δ θ m(x t n,θ) =1 v(x t n,θ) n δ θ v(x t n,θ) =1 v 2 (X t n,θ) θ [ ] X t n m(x t n,θ) [ ] (X t n m(x t n,θ)) 2 v(x t n,θ). E θ [M n (θ) M (θ) F ] = δ θm(x t n,θ) v(x t n,θ) E θ[(x t n n m(x t n,θ) X t n ] = δ θ v(x t n,θ) v 2 (X t n,θ) E θ[(x t n n m(x t n,θ)) 2 v(x t n,θ) X t n ], M n (θ) F n -. M n (θ). U n (θ) 179

,, M n (θ) = + 1 2 n δ θ m(x t n,θ) =1 v(x t n,θ) n δ θ v(x t n,θ) =1 v 2 (X t n,θ) θ [ ] X t n m(x t n,θ) [ ] (X t n m(x t n,θ)) 2 v(x t n,θ).

13 l n (θ), M n (θ) S n (θ). m(x, θ) v(x, θ) SDE, U n (θ) M n (θ).,,, M-., m(x, θ) v(x, θ)., 2 SDE m(x, θ) v(x, θ). SDE,,,., SDE,. 2.6, (2.7)-(2.8)., 2.3. Z t n := X (n) t n, Z t n = Z t n + b(z t n,α)h n + σ(z t n,β)(w t n w t n ), Z = x., w t n w t n, h n, L(Z t n Z t n = z), L(Z t n Z t n = z) N(z + b(z,α)h n,σ 2 (z,β)h n ).,., A2. E θ [X t n X t n ] = X t n + h n b(x t n,α)+o p (h 2 n), V θ [X t n X t n ] = h n σ 2 (X t n,α)+o p (h 2 n).,, (2.7)-(2.8) h n., g n. n g n (θ) = g(x t n,x t n,θ), (2.9) =1 g(x, y, θ) = 1 2 log(σ2 (x, β)) (y x h nb(x, α)) 2. (2.1) 2h n σ 2 (x, β) 18

, w t n w t n, h n, L(Z t n Z t n = z), L(Z t n Z t n = z) N(z + b(z,α)h n,σ 2 (z,β)h n ).,., A2.

14 , (2.7)-(2.8), (2.1) log(2πh nσ 2 (x, β)), θ, (2.1). ˆθ n =(ˆα n, ˆβ n ) g n (ˆθ n )=supg n (θ) θ., Kessler [14]. 1. A1-A3, nh 2 n, ( ) nhn (ˆα n α ) d N(,I 1 (θ )). n( ˆβn β ) g n, h n nh n, 1, nh 2 n., (2.7)-(2.8) h n. 1 3 (nh 2 n ), 4-5. nh 2 n,. Kessler [14] Ito-Taylor,,. 1,.,., (iii),. 1 ˆθ n,,. Gobet [8],, u, v R, P θ, l n (α + u/ nh n,β + v/ n) l n (α,β ) d ( u v ) Z 1 2 ( ) u I(θ ) v., Z, I(θ ) 2. (Local Asymptotic Normality, LAN). I(θ ), I(θ ) 1. 1 ˆθ n. ( u v ) 181

1 ˆθ n,,. Gobet [8],, u, v R, P θ, l n (α + u/ nh n,β + v/ n) l n (α,β ) d ( u v ) Z 1 2 ( ) u I(θ ) v., Z, I(θ ) 2.

15 3, 1. dx ε t = b(x ε t,α)dt + εσ(x ε t,β)dw t,t [,T], ε (, 1], X ε = x, (3.1) ε, SDE (2.1). ε, SDE (3.1) X ε t dx t = b(x t,α)dt, t [,T], X = x X t. SDE (2.1), SDE (3.1)., SDE (3.1) Xt ε., (Yoshida [33], Kunitomo and Taahashi [16])). 2,. Xt SDE (3.1) ε = : dxt = b(xt,α )dt, X = x. C,l (R Θ α ) f : (i)f(x, α) R Θ α, x, n, C>, x, sup α x n f(x, α) C(1 + x ) C. (ii) n, x n f(x, α) α l, ν =, 1,...,l, C>, x sup α δα ν x n f(x, α) C(1 + x ) C. 2 2 (Fisher ) I(θ ) ( ) I b (θ ) I(θ )= I σ (θ )., I b (θ ) = I σ (θ ) = 2 ( ) δα b(xs 2,α ) ds, σ(xs,β ) ( ) δβ σ(xs 2,β ) ds. σ(xs,β ) 182

1) ε = : dxt = b(xt,α )dt, X = x. C,l (R Θ α ) f : (i)f(x, α) R Θ α, x, n, C>, x, sup α x n f(x, α) C(1 + x ) C. (ii) n, x n f(x, α) α l, ν =, 1,.

16 11. ( ) dxt ε = (Xt ε β +(Xt ε ) α)dt + ε 2 1+(Xt ε ) dw t, t [,T], ε (, 1], X ε 2 = x., α, β >. ε =,, X t = x e t + α(1 e t ). 7-2 x =1.5, α =2, β =1, T =5, ε =.25, ε =.15, ε =.5, ε =, h n =1/1 Xt ε. ε, Xt ε, X t ( ) : , [,T], T =1., [, 1]., X ε n = {Xt ε } =,1,...,n, t = /n. X ε n α β., ε n. 2 (iii), t n = h n, h n, t n n = nh n, (i). 183

5, ε =, h n =1/1 Xt ε. ε, Xt ε, X t ( ). 2.2 7-2: 2 1.8 1.6 1.4 1 2 3 4 5, [,T], T =1., [, 1].

17 3.1 SDE (3.1),., T =1. B1 (i) C>, x, y R,θ Θ, b(x, α) b(y, α) + σ(x, β) σ(y, β) C x y. (ii) b(x, α) C,3 (R Θ α ). (iii) σ(x, β) C,3 (R Θ β ),, inf x,β σ 2 (x, β) >. B2 (i) I(θ ). (ii) α α t [, 1], b(x t,α) b(x t,α ). β β t [, 1], σ 2 (X t,β) σ 2 (X t,β ).,. { } G ε,n (θ) = 1 n log σ 2 (Xt ε 2,β)+ n (Xt ε Xt ε 1 n b(xε t,α)) 2. ε 2 σ 2 (Xt ε β) =1 2.6 (2.9)-(2.1). ˆθ ε,n =(ˆα ε,n, ˆβ ε,n ) G ε,n (ˆθ ε,n )=supg ε,n (θ). θ Θ., ˆθ ε,n (Sørensen and Uchida [28])). 2. B1-B2, (ε n) 1 = O(1), ( ε 1 ) (ˆα ε,n α ) d N (, I(θ ) 1). n( ˆβε,n β ) (ε n) 1, ˆβ ε,n ˆα ε,n., ε n, ε n,, ε := ε n (n ),,. 2, n ε, n := n ε (ε ). 184

ε 2 σ 2 (Xt ε β) =1 2.6 (2.9)-(2.1). ˆθ ε,n =(ˆα ε,n, ˆβ ε,n ) G ε,n (ˆθ ε,n )=supg ε,n (θ). θ Θ., ˆθ ε,n (Sørensen and Uchida [28])). 2. B1-B2, (ε n) 1 = O(1), ( ε 1 ) (ˆα ε,n α ) d N (, I(θ ) 1).

18 4,,, Praasa Rao [24, 25]., Genon-Catalot and Jacod [5, 6] (i) β,., (i), α,. Bibby and Sørensen [1] (ii). α β,. Uchida [29],,. Uchida [3], α M-,.,,., Kutoyants [17] Ibragimov and Has minsii [1], Küchler and Sørensen [2]. 1, Kutoyants [18, 19], Praasa Rao [25].,.,.,, [35], Ieda and Watanabe [11], Karatzas and Shreve [13], [3], [23],, Freidlin and Wentzell [2],, Jacod and Shiryaev [12],, Kloeden and Platen [15].,, ( [9]).,, Shimizu and Yoshida [27] Masuda [22],., Yoshida [32], 185

1, Kutoyants [18, 19], Praasa Rao [25].,.,.,, [35], Ieda and Watanabe [11], Karatzas and Shreve [13], [3], [23],,

19 ,,., [34], Saamoto and Yoshida [26]., 1,,.,. 5,. ( [35], Ieda and Watanabe [11], Karatzas and Shreve [13], [3], [23]),,. 5.1 Wiener 1. (Ω, F,P). (i) {X t } t 1. (ii) 1 {X t } t. P ({ω Ω [, ) t X t (ω) R }) =1. 2. (Ω, F,P). {F t } t, (i) t, F t σ-. (ii) s<t, F s F t F. (Ω, F,P; {F t } t ).,. 3. {X t } t 1. (i) t, X t F t -, {X t } t F t. 186

20 (ii) t, (s, ω) [,t] Ω X s (ω) R B([,t]) F t, {X t } t F t {w t } t (i)-(iv), 1 F t - Wiener. (i) {w t } t F t. (ii) P (w =)=1. (iii) t>s, w t w s F s,, t s 1. (iv) {w t } t F t - Wiener {w t } t,. (i) p =1, 2,..., E[w 2p t ]=(2p 1)!!t p, E[w 2p 1 t ]=. (ii) a.s. ω, w t (ω) t,. (iii) w t (ω) t 2. (iv) a.s. ω, w t (ω) t. 5. p 1. 1 X = {X t } t F t - L p (i)-(iv). (i) t, E[ X t p ] <. (ii) X = {X t } t F t. (iii) t>s, E[X t F s ]=X s a.s. (iv) {X t } t. 1 F t - Wiener {w t } t F t - L p. 187

!t p, E[w 2p 1 t ]=. (ii) a.s. ω, w t (ω) t,. (iii) w t (ω) t 2. (iv) a.s. ω, w t (ω) t. 5. p 1.

21 5.2 (Ω, F,P; {F t } t ) (i)-(iii),. (i)(ω, F,P),, M N F P (N) = M F. (ii) F t,, t F t = s>t F s. (iii) F P -,, N := {A Ω B F A B,P(B) =} F., (Ω, F,P; {F t } t )), 1 F t - Wiener {w t } t. w t a.s. ω, t ( 3-(iv)), f t (ω)dw t (ω) (5.1) ω Lebesgue-Stieltjes., Wiener w, L 2 loc (F t) f Wiener w (5.1)., L 2 loc (F t)={{f t } t 1 F t, T, P( } T f 2 t (ω)dt < ) =1., [35] [23]. L 2 (F t )={{f t } t 1 F t, T, E[ } T f 2 t (ω)dt] <. L 2 (F t ) L 2 loc (F t). 4. f L 2 (F t ). (i) { t f sdw s }t F t - L 2. (ii) T, [ ] E f t dw t [ ( ) 2 ] E f t dw t =, [ = E ] ft 2 dt. L 1 loc (F t)={{f t } t 1 F t, T, P ( } T f t(ω) dt < ) =1. 188

22 5 ( ). X F -, b L 1 loc (F t), σ L 2 loc (F t). 1 ( ) X T = X + b t dt +., f C 2 (R), f(x T ) = f(x )+ + f x (X t)b t dt f x (X t)σ t dw t σ t dw t 2 f x 2 (X t)σ 2 t dt., (t, x) f(t, x) x 2, t 1, (t, x),. f(t,x T ) = f(,x )+ + + f t (t, X t)dt f x (t, X t)b t dt f x (t, X t)σ t dw t 2 f x 2 (t, X t)σ 2 t dt 5.3 (Ω, F,P; {F t } t ), 1 F t - Wiener {w t } t. b, σ R R Borel. 6. X t (ω) :[, ) Ω R x dx t = b(x t )dt + σ(x t )dw t, X = x (5.2),. (i) X = {X t } t F t, T ( { P b(xt ) + σ 2 (X t ) } ) dt < =1. 189

23 (ii) P (X = x )=1, (iii) X t = X + t b(x s)dt + t σ(x s)dw t, t (Lipschitz) L>, x, y R b(x) b(y) + σ(x) σ(y) L x y., (5.2) X = {X t } t., : X Y (5.2), P (X t = Y t, t ) = 1. [1] Bibby, B. M. and Sørensen, M. (1995). Martingale estimating functions for discretely observed diffusion processes. Bernoulli 1, [2] Freidlin M. I. and Wentzell A. D. (1998). Random perturbations of dynamical systems, second edition. Springer-Verlag, New Yor. [3] (1997)... [4] Genon-Catalot, V. (199). Maximum contrast estimation for diffusion processes from discrete observations. Statistics 21, [5] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Statist. 29, [6] Genon-Catalot, V. and Jacod, J. (1994). Estimation of the diffusion coefficient for diffusion processes: random sampling. Scand. J. Statist. 21, [7] Gobet, E. (21). Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7, [8] Gobet, E. (22). LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincare Probab. Statist. 38, [9] (28).. 21 I:, 1,. 19

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untitled 3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t)