In this paper, we study a limit theorem for stochastic processes Xn t given ...... To estimate Jn. 4 , using Taylor's theorem (Theorem 1.43 in [12]), we have. E n[F.
J. Math. Sci. Univ. Tokyo 12 (2005), 233–288.
A Limit Theorem for Solutions of Some Functional Stochastic Difference Equations By Takashi Kato Abstract. In this paper, we study a limit theorem for solutions of some functional stochastic difference equations under strong mixing conditions and some dimensional conditions. This work is an extension of the work of Hisao Watanabe.
1.
Introduction and Main Results
Diffusion approximations for certain stochastic difference equations or stochastic ordinary differential equations have been discussed in several papers. [9] [15], [16] and [17] treated such problem and derived the weak limit of appropriately scaled and interpolated process, which was given by the solution of a stochastic difference equation as a diffusion process. Concerning this, [5], [6], [10], [11] and many other papers dealt with weak convergence of the solution of a stochastic ordinary differential equation. In this paper, we study a limit theorem for stochastic processes Xtn given by the following functional stochastic difference equations (1.1)
1 1 n n X(k+1)/n − Xk/n = √ Fkn (X n , ω) + Gnk (X n , ω) n n
and by linear interpolation as (1.2)
n n + (nt − k)X(k+1)/n Xtn = (1 − nt + k)Xk/n
for k/n < t < (k + 1)/n, and (1.3)
X0n = x0 ∈ Rd .
Here Fkn and Gnk are d dimensional random functions on C([0, ∞); Rd ), the space of continuous functions from [0, ∞) to Rd , such that Fkn has mean zero. 1991 Mathematics Subject Classification. 233
Primary 60F05; Secondary 60B10, 39A12.
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Takashi Kato
Under certain assumptions for Fkn and Gnk , we show that the distribution of X n converges weakly to the solution of a martingale problem corresponding to functional coefficients. The methods of the proof are based on [5] and [16]. However, we cannot use mixing inequalities in these papers, since the dimension of parameter space C([0, ∞); Rd ) is infinite. We show another version of mixing inequalities by assuming certain dimensional conditions for the set of random variables Fkn (w) and Gnk (w), which may look artificial but we give sufficient conditions for this assumption later. The author thanks Professor Shigeo Kusuoka for a lot of precious advice and discussions. Let (Ωn , F n , P n ), n ∈ N = {1, 2, 3, . . . }, be complete probability spaces. d Let Fkn (w, ω) = (Fkn,i (w, ω))di=1 and Gnk (w, ω) = (Gn,i k (w, ω))i=1 : C([0, ∞); d n d R ) × Ω −→ R , k ∈ Z+ = {0, 1, 2, . . . }, be random functions. Let Bt be the σ-algebra of C([0, ∞); Rd ) given by Bt = σ(w(s) ; s ≤ t). We introduce the following conditions. n [A1] Fkn,i and Gn,i k are measurable with respect to Bk/n ⊗ F .
By [A1], we can regard Fkn,i and Gn,i k as random functions defined on d the Banach space C([0, k/n]; R ). [A2] Fkn,i (w, ω) (respectively, Gn,i k (w, ω)) is twice (respectively, once) continuously Fr´echet differentiable in w for P n -almost surely ω. We denote by Lm T the space of real valued continuous m-multilinear operators on C([0, T ]; Rd ) and denote by | · |Lm its norm. Then the m-th T m n,i Fr´echet derivative ∇ Fk (w) : (w1 , . . . , wm ) −→ ∇m Fkn,i (w; w1 , . . . , wm ) m n,i is regarded as the element of Lm k/n for each w (and so is ∇ Gk (w)). For m = 0, L0T = R and ∇0 Fkn,i (w) = Fln,i (w). Let p0 > 3 and γ0 > 0. We assume the moment conditions with respect to p0 and the dimensional conditions with respect to γ0 as [A3] and [A4].
Limits of Functional Stochastic Difference Equations
235
[A3] For each M > 0, there exists a constant C(M ) > 0 such that 2
(1.4)
E
n
E
n
p sup ∇m Fkn,i (w)L0m ≤ C(M )
|w|∞ ≤M
m=0
k/n
and 1
(1.5)
m=0
p0 m ≤ C(M ) (w) sup ∇m Gn,i k L
|w|∞ ≤M
k/n
for any n ∈ N and k ∈ Z+ , where E n [·] denotes the expectation under the probability measure P n and |w|∞ = sup |w(t)|. t≥0
d denote the set of w ∈ C([0, ∞); Rd ) such that |w|∞ ≤ M . For a Let CM random function U : C([0, ∞); Rd ) × Ωn −→ R and ε > 0, let Nn (ε, M ; U ) be the smallest integer m such that there exist sets S1 , . . . , Sm which satisfy m d CM = Si and i=1
E
n
max
sup |U (x) − U (y)|
i=1,... ,m x,y∈Si
p0
1/p0
< ε.
[A4] (1.6)
sup sup εγ0 Nn (ε, M ; Fkn,i ) < ∞, n,k ε>0
(1.7)
sup sup sup εγ0 Nn (ε, M ; ∇Fkn,i (· ; Iln ej )) < ∞, n,k l≤k ε>0
(1.8)
n eν )) < ∞, sup sup sup εγ0 Nn (ε, M ; ∇2 Fkn,i (· ; Iln ej , Im n,k l,m≤k ε>0
(1.9)
sup sup εγ0 Nn (ε, M ; Gn,i k )0
and (1.10)
n sup sup sup εγ0 Nn (ε, M ; ∇Gn,i k (· ; Il ej )) < ∞ n,k l≤k ε>0
for each M > 0 and i, j, ν = 1, . . . d, where ej ∈ Rd denotes the unit vector j along the j-th axis, i.e. ej = (0, . . . , 0, ˇ1, 0, . . . , 0), and the function Iln :
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Takashi Kato
[0, ∞) −→ R is given by l 0 if 0 ≤ t < n l+1 l n Il (t) = ≤t< nt − l if n n l + 1 1 if t ≥ . n [A5] Let
n,i n d = σ Fm (w), Gn,i Fk,l m (w) ; i = 1, . . . , d, k ≤ m ≤ l, w ∈ C([0, ∞); R ) and n n αk = sup sup sup{|P n (A ∩ B) − P n (A)P n (B)| ; A ∈ F0,l , B ∈ Fk+l,∞ }. n
l
Then ∞
(1.11)
αk0 < ∞,
k=1
where (0 =
1 p0 and s0 = . 2s0 + 4γ0 p0 − 3
[A6] E n [Fkn,i (w)] = 0. We denote by Kd the family of a compact set K of C([0, ∞); Rd ) such that sup |w|∞ < ∞. w∈K
[A7] Let n,i n,j n an,ij 0 (k, w) = E [Fk (w)Fk (w)], n,i n bn,i 0 (k, w) = E [Gk (w)], ∞ k
n,j n,i n Fk (w) , An,ij (k, w) = E Fk+l w · ∧ n
B n,ij (k, w) =
l=1 ∞ l=1
E
n
k n,i ∇Fk+l w · ∧ ; Ikn ej Fkn,j (w) n
Limits of Functional Stochastic Difference Equations
237
for k ∈ Z+ and w ∈ C([0, ∞); Rd ), where a ∧ b = min{a, b}. The following limits exist uniformly on any K ∈ Kd for each t ≥ 0 : (1.12)
aij 0 (t, w) =
(1.13)
bi0 (t, w) =
(1.14)
Aij (t, w) =
(1.15)
B ij (t, w) =
lim an,ij ([nt], w), n→∞ 0 lim bn,i ([nt], w), n→∞ 0 lim An,ij ([nt], w),
n→∞
lim B n,ij ([nt], w),
n→∞
where [x] denotes the greatest integer less than or equal to x. [A8] Define a(t, w) = (aij (t, w))di,j=1 and b(t, w) = (bi (t, w))di=1 by ij ji aij (t, w) = aij 0 (t, w) + A (t, w) + A (t, w)
and bi (t, w) = bi0 (t, w) +
d
B ij (t, w).
j=1
For each T > 0, there exists a positive constant C(T ) such that
(1.16) |aij (t, w)| ≤ C(T ) 1 + sup |w(s)|2 0≤s≤t
and (1.17)
|bi (t, w)| ≤ C(T ) 1 + sup |w(s)| 0≤s≤t
for t ∈ [0, T ] and w ∈ C([0, ∞); Rd ). [A9] Let d d 1 ij ∂2 ∂ f (t, w) = a (t, w) i j f (w(t)) + bi (t, w) i f (w(t)) 2 ∂x ∂x ∂x i,j=1
i=1
for f ∈ C 2 (Rd ). The martingale problem associated with the generator and initial value x0 has a unique solution Q on C([0, ∞); Rd ). We will introduce the sufficient conditions for [A4] and [A9] in Section 5.
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Takashi Kato
Define the stochastic process Xtn = (Xtn,i )di=1 by (1.1), (1.2) and (1.3). Let Qn be the probability measure induced by X n on C([0, ∞); Rd ). Theorem 1. Assume [A1] − [A9]. Then Qn converges weakly to Q on C([0, ∞); Rd ). Let us give some remarks on Theorem 1. (i) In fact, using the arguments in [16], we can prove Theorem 1 without assuming the condition (1.10). (ii) We can replace the assumption [A5] with [A5 ] For each M > 0 ∞
(1.18)
αk (M )0 < ∞,
k=1
where n n,i Fk,l (M ) = σ(Fm (w), Gn,i m (w) ; i = 1, . . . , d, k ≤ m ≤ l, |w|∞ ≤ M )
and αk (M ) = sup sup sup{|P n (A ∩ B) − P n (A)P n (B)| ; n
l n n A ∈ F0,l (M ), B ∈ Fk+l,∞ (M )}.
The proof needs no change. (iii) Assuming the following uniform mixing condition [A5 ] instead of [A5], we can remove the dimensional condition [A4] : [A5 ] It holds that ∞
(1.19)
φk2 < ∞,
k=1
where (2 =
p0 − 2 and 2p0
P n (A ∩ B) n φk = sup sup sup (B) − P ; n (A) P n l
n n A ∈ F0,l , B ∈ Fk+l,∞ , P n (A) > 0 .
Limits of Functional Stochastic Difference Equations
239
Next we provide another version of Theorem 1. We introduce the following conditions. [B4] For some γ1 > 0, (1.6)−(1.10) hold with log Nn instead of Nn . 1 such that [B5] Let αk be as in [A5]. Then there exists (1 ∈ 0, 2γ1 ∞
(1.20)
k=1
1 1 < ∞. log(1/αk )
Theorem 2. Assume [A1] − [A3], [B4], [B5] and [A6] − [A9]. Then Q converges weakly to Q on C([0, ∞); Rd ). n
2.
Mixing Inequalities
In this section we prepare some inequalities for strong mixing coefficients. Let (Ω, F, P ) be a probability space and A, B, C ⊂ F be sub σ-algebras. Define α(A, B) by α(A, B) = sup{ |P (A ∩ B) − P (A)P (B)| ; A ∈ A, B ∈ B }. The following lemma is shown in the proof of Theorem 17.2.2 in [4]. 1 1 1 + + = 1, X be an p q r A-measurable random variable and Y be a B-measurable random variable. Then E[XY ] − E[X] E[Y ] ≤ 8 E[|X|p ]1/p E[|Y |q ]1/q α(A, B)1/r . (2.1) Lemma 1. Let 1 ≤ p, q, r ≤ ∞ be such that
Let (S, d) be a metric space, ε, p > 0 and U : S×Ω −→ R be a continuous random function. We say that a family of sets (Si )m i=1 is an (ε, p, U )-net of m S if S = Si and i=1
E
max
sup |U (x) − U (y)|p
i=1,... ,m x,y∈Si
1/p
< ε.
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Takashi Kato
We denote the minimum of cardinals of (ε, p, U )-nets by N (ε, p; U ). 1 1 + < 1 and U : S × Ω −→ p q R be a continuous random function such that U (x) is A-measurable and E[U (x)] = 0 for each x ∈ S, and X : Ω −→ S, V : Ω −→ R be B-measurable random variables. Then for any ε > 0 Lemma 2. Let 1 < p, q < ∞ be such that
(2.2)
where
E[U (X)V ] ≤ 8 E[sup |U (x)|p ]1/p + 1 x∈S × E[|V |q ]1/q ε + ε1−r N (ε, p; U )α(A, B) , 1 1 1 =1− − . r p q
Proof. We may assume that the right-hand side of (2.2) is finite and α(A, B) > 0. Set Nε = N (ε, p; U ) and U ∗ = sup |U (x)|. Let δ = p/r, δ˜ = x∈S
q/r, ˜
I = E[|U ∗ |p ]1/p ε−1/δ , J = E[|V |q ]1/q ε−1/δ and UI (x) = U (x)1{|U ∗ |≤I} , VJ = V 1{|V |≤J} . Then we have (2.3)
1 1 + = r − 1. δ δ˜
ε Let (Si )N i=1 be an (ε, p, U )-net. We may assume that all Si are disjoint ˜ : Ω −→ and not empty. Take any xi ∈ Si , and define the random variable X S by Nε ˜ X(ω) = xi 1Ωi (ω),
i=1
where Ωi = {X ∈ Si }. Then it follows that ˜ ˜ − UI (X))V ˜ ] + E[(U (X) (2.4) E[U (X)V ] ≤ E[(U (X) − U (X))V ] ˜ ˜ J ]. + E[UI (X)(V − VJ )] + E[UI (X)V
Limits of Functional Stochastic Difference Equations
˜ we have By the definition of X, ˜ E[(U (X) − U (X))V (2.5) ] ≤ E max sup |U (x) − U (y)| · |V | ≤ E
i=1,... ,Nε x,y∈Si
sup |U (x) − U (y)|p
max
1/p
i=1,... ,Nε x,y∈Si
q 1/q E[|V | ]
≤ ε E[|V |q ]1/q . By the Chebyshev inequality and the H¨ older inequality, we have (2.6)
1 ˜ − UI (X))V ˜ E[(U (X) ] ≤ δ E |U ∗ |1+δ |V | I 1 ∗ p (1+δ)/p q 1/q ∗ p 1/p q 1/q ≤ E[|U | ] E[|V | ] = E[|U | ] E[|V | ] ε. Iδ
Similarly we obtain ˜ E[UI (X)(V (2.7) − VJ )] ≤ E[|U ∗ |p ]1/p E[|V |q ]1/q ε. ¯I (x) = E[UI (x)] and Set U ˜ J ] ≤ (2.8) E[UI (X)V
˜I (x) = UI (x) − U ¯I (x). Then it follows that U ¯I (X)V ˜I (X)V ˜ J ] + E[U ˜ J ] E[U
¯I (x)| E[|V |q ]1/q + ≤ sup |U x∈S
Nε ˜I (xi )VJ 1Ω ]. E[U i i=1
Since E[U (x)] = 0, we have (2.9)
¯I (x)| = E[UI (x) − U (x)] ≤ 1 E[|U ∗ |1+δ ] = E[|U ∗ |p ]1/p ε. |U Iδ
By Lemma 1 and (2.3), we get (2.10)
Nε ˜I (xi )VJ 1Ω ] ≤ 8Nε IJα(A, B) E[U i i=1
= 8 E[|U ∗ |p ]1/p E[|V |q ]1/q ε1−r Nε α(A, B). By (2.4)-(2.10), we obtain the assertion.
241
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Takashi Kato
1 1 + < 1 and U : S × Ω −→ p q R be a continuous random function such that U (x) is A-measurable and E[U (x)] = 0 for each x ∈ S, and X : Ω −→ S, V : Ω −→ R be B-measurable random variables. Suppose that there exist positive constants C0 and γ such that Lemma 3. Let 1 < p, q < ∞ be such that
sup εγ N (ε, p; U ) ≤ C0 .
(2.11)
ε>0
Then it holds that
E[U (X)V ] ≤ 16(C0 + 1) E[sup |U (x)|p ]1/p + 1 (2.12) x∈S
× E[|V | ]
q 1/q
where ( =
α(A, B) ,
1 1 1 1 and = 1 − − . r+γ r p q
Proof. By Lemma 2, we get
E[U (X)V ] ≤ 8(C0 + 1) E[sup |U (x)|p ]1/p + 1 x∈S
× E[|V | ]
q 1/q
ε + ε1−r−γ α(A, B) .
The assertion now follows by taking ε = α(A, B) . We denote by A ∨ B the smallest σ-algebra which includes both A and B. The following lemma is obtained by Lemma 3 and the arguments in the proof of Lemma 2 in [5]. 1 1 1 + + < 1. Let p q r U, V : S × Ω −→ R be continuous random functions such that U (x) and V (x) are A and B-measurable respectively and E[U (x)] = 0 for each x ∈ S, and X : Ω −→ S, Z : Ω −→ R be C-measurable random variables. Suppose that there exist positive constants C0 , u∗ , v ∗ and γ such that (2.13) sup εγ N (ε, p; U ) + N (ε, q; V ) ≤ C0 , Lemma 4. Let 1 < p, q, r < ∞ be such that
ε>0
(2.14)
p 1/p ∗ E[sup |U (x)| ] ≤ u x∈S
Limits of Functional Stochastic Difference Equations
243
and q 1/q ∗ E[sup |V (x)| ] ≤ v .
(2.15)
x∈S
Then there exists a constant C > 0 depending only on C0 , u∗ , v ∗ and γ such that E[Ξ(X)Z] ≤ C E[|Z|r ]1/r α(A ∨ B, C) α(A, B ∨ C) , (2.16) where Ξ(x) = U (x)V (x)−E[U (x)V (x)], ( =
Proof. Set ε˜ =
2(u∗
1 1 1 1 1 and = 1− − − . 2s + 4γ s p q r
ε 1 1 1 . Let t ≥ 1 be such that = + . Then ∗ +v ) t p q
we have N (ε, t; Ξ) ≤ N (˜ ε, p; U )N (˜ ε, q; V ).
(2.17)
N (˜ ε,p,U ) N (˜ ε,q,V ) and (S˜j )j=1 be (˜ ε, p, U )-net and (˜ ε, p, U )-net Indeed, if we let (Si )i=1 respectively, then the H¨ older inequality implies 1/t t E max sup |Ξ(x) − Ξ(y)| i,j
x,y∈Si ∩S˜j
≤ 2 E sup |U (x)|t max sup |V (x) − V (y)|t j
x∈S
1/t
x,y∈S˜j
+ E max sup |U (x) − U (y)|t sup |V (x)|t i
x,y∈Si
j
x,y∈S˜j
1/t
x∈S
1/q ≤ 2 u∗ E max sup |V (x) − V (y)|q
+ E max sup |U (x) − U (y)|p i
∗
1/p
v∗
x,y∈Si
∗
ε = ε. ≤ 2(u + v )˜ Thus (Si ∩ S˜j )i=1,... ,N (˜ε,p;U ),j=1,... ,N (˜ε,q;V ) is an (ε, t, Ξ)-net. This implies (2.17). So we get (2.18)
N (ε, t; Ξ) ≤ 22γ (u∗ + v ∗ )2γ C02 ε−2γ .
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Takashi Kato
Then, using Lemma 3 with Ξ substituted for U , we have
(2.19) E[Ξ(X)Z] ≤ C1 E[sup |Ξ(x)|t ]1/t + 1 E[|Z|r ]1/r α(A ∨ B, C) x∈S ∗ ∗
≤ 2C1 (u v + 1) E[|Z|r ]1/r α(A ∨ B, C)2
for some C1 > 0 depending only on C0 , u∗ , v ∗ and γ > 0. On the other hand, using Lemma 3 with V (X)Z substituted for V , we have (2.20) E[U (X)V (X)Z] ≤ C2 (u∗ + 1) E[|V (X)Z|t ]1/t α(A, B ∨ C)
≤ C2 (u∗ + 1)v ∗ E[|Z|r ]1/r α(A, B ∨ C)2 . for some C2 > 0 depending only on C0 and γ > 0, where 1 . s+γ Set W (x) = E[U (x)V (x)]. By Lemma 1, we see
1 1 1 = + and t q r
( =
|W (x)| ≤ 8u∗ v ∗ α(A, B)1−1/t ≤ 8u∗ v ∗ α(A, B ∨ C)2
for each x ∈ S. Thus E[W (X)Z] ≤ 8u∗ v ∗ E[|Z|r ]1/r α(A, B ∨ C)2 . (2.21) By (2.19), (2.20) and (2.21), it follows that E[Ξ(X)Z] = E[Ξ(X)Z]1/2 E[Ξ(X)Z]1/2
≤ C3 E[|Z|r ]1/r α(A ∨ B, C) α(A, B ∨ C)
for some C3 > 0 depending only on C0 , u∗ , v ∗ and γ > 0. This implies the assertion. 3.
Proof of Theorem 1 Let ϕM ∈ C ∞ (Rd ; R) be such that 0 ≤ ϕM ≤ 1, ϕM (x) =
1 0
if |x| ≤ M/2 if |x| ≥ M,
Limits of Functional Stochastic Difference Equations
245
and the gradient of ϕM (x) is bounded uniformly in x ∈ Rd and M ≥ 1. Define the truncated functions Fkn,M (w) = (Fkn,M,i (w))di=1 and Gn,M (w) = k n,M,i d (w))i=1 by (Gk (w) = ϕM (w(k/n))Gnk (w). Fkn,M (w) = ϕM (w(k/n))Fkn (w), Gn,M k We also define the stochastic process Xtn,M = (Xtn,M,i )di=1 by (1.1) and (1.2) for which Fkn and Gnk are replaced by Fkn,M and Gn,M . k n,M,i To make notations simple, we set Hk (w) = Fkn,M,i (w) + 1 √ Gn,M,i (w). Then Xtn,M,i satisfies the following equation n k (3.1)
1 n,M,i n,M,i − Xk/n = √ Hkn,M (X n,M ). X(k+1)/n n
Proposition 1. For each ω ∈ Ωn , if |Xtn,M (ω)| ≤ M , then |Xsn,M (ω)| ≤ M for any s ∈ [0, t]. Proof. We prove the contraposition of the assertion. Suppose that n,M |Xsn,M | > M holds for some s ∈ [0, t]. Let k = [ns]. If |Xk/n | > M , we n,M | ≤ M. have |Xtn,M | = |Xsn,M | > M obviously. So we may suppose |Xk/n n,M n,M Then we see |X(k+1)/n | > M . Indeed, if |X(k+1)/n | ≤ M , then |Xsn,M | ≤
M holds by the convexity of the set {x ∈ Rd ; |x| ≤ M }, and this contradicts n,M the supposition. So Xtn,M is in {uXsn,M + (1 − u)X(k+1)/n ; 0 ≤ u ≤ 1} ⊂ n,M n,M ; u ≥ 1}. Since |Xk/n | ≤ M and |Xsn,M | > M hold, {uXsn,M + (1 − u)Xk/n n,M | > M for each u ≥ 1. Thus |Xtn,M | > M we have |uXsn,M + (1 − u)Xk/n holds and we obtain the assertion.
By Proposition 1, the assumption [A3] and the definition of Xtn,M , we see n that Xtn,M is F0,[nt] -measurable and that there exists a constant C(M ) > 0 such that (3.2)
2 m=0
m n,M,i n,M p0 ∇ ≤ C(M ) F (X ) E k Lm n
k/n
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Takashi Kato
and 1
(3.3)
E
n
∇m Gn,M,i (X n,M )p0m ≤ C(M ) k L k/n
m=0
for n ∈ N and k ∈ Z+ . Let
n,M n,M n,M Ykn,M (u, t) = Xt∧(k/n) + u Xt∧((k+1)/n) − Xt∧(k/n) , u ∈ [0, 1]. Easily we have
(3.4)
k n,M X if t ≤ t n k k+1 n,M n,M Yk (u, t) = Xk/n+u(t−k/n) if 0 such that Proposition 2. Let 1 < q < ∞ be such that
(3.5)
sup εγ Nn (ε, M ; U ) ≤ C0 . ε>0
Then there exists a constant C > 0 depending only on M and C0 such that for all l ≤ k, u ∈ [0, 1] and β = (β 1 , . . . , β d ) ∈ Zd+ with |β| = β 1 + · · · + β d ≤ 2 n M n,M E [U (Y (3.6) (u, ·))V ] β l 0 , ≤ C(E n [ sup |U (w)|p0 ]1/p0 + 1) E n [|V |q ]1/q αk−l |w|∞ ≤M
where UβM (w) = Dβ ϕM (w(k/n))U (w) and Dβ =
∂ |β| . ∂xβ 1 · · · ∂xβ d
Limits of Functional Stochastic Difference Equations
247
Proof. Define Yˆln,M (u, t) and Vˆ by n,M ≤M Yln,M (u, t) if X(l+u)/n n,M ˆ (u, t) = (3.7) Yl 0 otherwise and
Vˆ =
n,M ≤M if X(l+u)/n otherwise .
V 0
By (3.4) and Proposition 1, we see that |Yˆln,M (u, t)| ≤ M for all t ≥ 0 almost surely and (3.8)
n,M n,M n M (u, ·))V ] = E n [U (Yˆln,M (u, ·))Dβ ϕM (X(l+u)/n )Vˆ ]. E [Uβ (Yl
Using Lemma 3, we see that n E [U (Yˆ n,M (u, ·))Dβ ϕM (X n,M )Vˆ ] l (l+u)/n 1/p0
+1 ≤ 16(C0 + 1) E n sup |U (w)|p0 |w|∞ ≤M
0 n,M )Vˆ |q ]1/q αk−l , × E n [|Dβ ϕM (X(l+u)/n
1 1 1 1 and = 1− − . Since s0 ≤ 2s0 holds, which implies s0 + γ s0 p0 q (0 ≥ 2(0 , and Dβ ϕM is bounded uniformly in x, we have our assertion. where (0 =
Proposition 3. Let U, V : C([0, ∞); Rd ) × Ωn −→ R be such that n n U (w) and V (w) are Fk,k and Fl,l -measurable respectively and E n [U (w)] = 0 d n n for each w ∈ CM , and Z : Ω −→ R be an F0,m -measurable random variable. Suppose that there exists C0 = C0 (M ) > 0 such that (3.9) sup εγ Nn (ε, M ; U ) + εγ Nn (ε, M ; V ) ≤ C0 , ε>0
(3.10)
E
n
sup |U (w)|p0
1/p0
|w|∞ ≤M
and (3.11)
E
n
sup |V (w)|p0
|w|∞ ≤M
1/p0
≤ C0 .
≤ C0
248
Takashi Kato
Then there exists a constant C > 0 depending only on M and C0 such that for all m ≤ l ≤ k, u ∈ [0, 1] and β, β ∈ Zd+ with |β| + |β | ≤ 2 n M E [Ξ (Ymn,M (u, ·))Z] ≤ C E n [|Z|p0 ]1/p0 α0 α0 , β,β k−l l−m
where ΞM = Dβ ϕM (w(k/n))Dβ ϕM (w(l/n))Ξ(w), β,β (w) U (w)V (w) − E n [U (w)V (w)].
Ξ(w)
=
Proof. Define Zˆ by n,M ≤M Z if X(m+u)/n ˆ Z= 0 otherwise . Then we have (3.12)
n M n,M E [Ξ (Ym (u, ·))Z] n,M n,M ˆ = E n [Ξ(Yˆmn,M (u, ·))Dβ ϕM (X(m+u)/n )Dβ ϕM (X(m+u)/n )Z],
where Yˆmn,M (u, t) is given by (3.7). Using Lemma 4, we see that there exists C1 > 0 depending only on M and C0 such that n 2ˆ E [Ξ(Yˆmn,M (u, ·))ϕM (X n,M (m+u)/n ) Z] n,M n,M ˆ p0 ]1/p0 α0 α0 . ≤ C1 E n [|Dβ ϕM (X(m+u)/n )Dβ ϕM (X(m+u)/n )Z| k−l l−m
Then we have our assertion. Let Qn,M be the probability measure induced by X n,M on C([0, ∞); Rd ). Proposition 4. The family of measures (Qn,M )n is tight for each fixed M > |x0 |. Proof. Take any T > 0. Let 0 ≤ s < t < u ≤ T, 0 < δ0 < and set
p0 − 3 ∧1 2
J0n = E n [|Xun,M,i − Xtn,M,i |2 |Xtn,M,i − Xsn,M,i |1+δ0 ]. By the argument in [1], [5] and [16], it suffices to show that there exists a constant C0 = C0 (M, T ) > 0 which is independent of s, t, u and n such that (3.13)
J0n ≤ C0 |u − s|1+1/q0 ,
Limits of Functional Stochastic Difference Equations
249
p0 . 1 + δ0 First we consider the case of u − s < 1/n. In this case, it follows that [ns] + 1 = [nt] = [nu] or [ns] = [nt] = [nu] − 1. If [ns] + 1 = [nt] = [nu], by assumption [A3] and Proposition 1, we have
where q0 =
√ 2 n,M (3.14) J0n = E n n(u − t)H[nt] (X n,M ) 1 n,M (X n,M ) × √ (nt − [nt])H[nt] n 1+δ0 1 n,M (X n,M ) + √ (1 − ns + [ns])H[ns] n n,M,i n,M 2 √ (X ) = ( n)1−δ0 |u − s|2 E n H[nt] n,M,i × (nt − [nt])H[nt] (X n,M ) 2 n,M,i +(1 − ns + [ns])H[ns] (X n,M ) √ n,M,i ≤ ( n)1−δ0 |u − s|2 E n [|H[nt] (X n,M )|p0 ](3+δ0 )/p0 n,M,i +E n [|H[nt] (X n,M )|p0 ]2/p0 n,M,i ×E n [|H[ns] (X n,M )|p0 ](1+δ0 )/p0
√ ≤ C1 ( n)1−δ0 |u − s|2 ≤ C1 |u − s|(3+δ0 )/2 ≤ C2 |u − s|1+1/q0 for some C1 = C1 (M ) > 0 and C2 = C2 (M, T ) > 0. If [ns] = [nt] = [nu] − 1, the similar calculation gives us the following estimation J0n ≤ C3 |u − s|1+1/q0 for some C3 = C3 (M, T ) > 0. So the inequality (3.13) holds when u − s < 1/n. Next we consider the case of u − s ≥ 1/n. We will show that there exists a constant C4 = C4 (M, T ) > 0 such that (3.15)
n n,M,i − Xrn,M,i |2 Φ] ≤ C4 |u − s| E n [Φq0 ]1/q0 E [|Xv
n for each r, v ∈ [s, u] with r ≤ v and each F0,([nr]−1)∨0 -measurable nonnegative random variable Φ.
250
Takashi Kato
Since we have |Xvn,M,i − Xrn,M,i |2 n,M,i n,M,i 2 ≤ 3 |X([nv]+1)/n − Xvn,M,i |2 + |Xrn,M,i − X[nr]/n | [nv] 2
n,M,i n,M,i X(k+1)/n − Xk/n + k=[nr]
and the following equality (3.16)
k
xl
l=1
2
=
k
x2l
l=1
+2
k
xl (x1 + · · · + xl ), x1 , . . . , xk ∈ R,
l=1
it follows that n n,M,i − Xrn,M,i |2 Φ] ≤ 6(J1n + J2n + J3n + J4n + J5n ), E [|Xv
where n,M,i J1n = E n [|X([nv]+1)/n − Xvn,M,i |2 Φ], n,M,i 2 | Φ], J2n = E n [|Xrn,M,i − X[nr]/n
J3n
=
[nv] 1 n n,M,i n,M 2 (X ) Φ , E Hk n k=[nr]
J4n
=
[nv] 1 n n,M,i n,M n,M,i n,M,i √ (X )(Xk/n − X[nr]/n )Φ], E [Fk n k=[nr]
J5n
=
[nv] 1 n n,M,i n,M n,M,i n,M,i (X )(Xk/n − X[nr]/n )Φ]. E [Gk n k=[nr]
Since (3.17) (3.18)
2 1 + < 1, we have p0 q 0 1 n,M,i (X n,M )|p0 ]2/p0 E n [Φq0 ]1/q0 ([nv] + 1 − v)2 E n [|H[nv] n 1 ≤ C5 × E n [Φq0 ]1/q0 ≤ C5 |u − s| E n [Φq0 ]1/q0 n
J1n ≤
Limits of Functional Stochastic Difference Equations
251
for some C5 = C5 (M ) > 0. Similarly we have J2n ≤ C6 |u − s| E n [Φq0 ]1/q0
(3.19)
for some C6 = C6 (M ) > 0. We also have [nv] − [nr] + 1 n p0 1/p0 J3n ≤ C7 · E [Φ ] n
2 n q 1/q n q 1/q ≤ C7 |v − r| + E [Φ 0 ] 0 ≤ 3C7 |u − s| E [Φ 0 ] 0 n
(3.20)
for some C7 = C7 (M ) > 0. To estimate J4n , using Taylor’s theorem (Theorem 1.43 in [12]), we have n,M,i n,M,i n,M,i n (X n,M )(Xk/n − X[nr]/n )Φ] E [Fk k−1
=
E
l=[nr]
=
n
n,M,i n,M
n,M,i n,M,i Fk Φ (X·∧((l+1)/n) ) X(l+1)/n − Xl/n
n,M + E n Fkn,M,i (X·∧((l+1)/n) )
n,M n,M,i n,M,i ) Xl/n − X[nr]/n −Fkn,M,i (X·∧(l/n) Φ
k−1 1 n,(1) n,(2) n,(3) √ Λk,l + Λk,l + Λk,l , n l=[nr]
where n,(1)
Λk,l
n,(2)
Λk,l
n,M n,M = E n ϕM (X(l+1)/n )Fkn,i (X·∧((l+1)/n) )Hln,M,i (X n,M )Φ ,
=
d j=1
n,(3) Λk,l
=
E
n
E
n
0
d j=1
1
0
1
∂ ϕM (Yln,M (u, k/n))Fkn,i (Yln,M (u, ·)) ∂xj
n,M,i n,M,i Φ du, − X[nr]/n ×Hln,M,j (X n,M ) Xl/n ϕM (Yln,M (u, k/n))∇Fkn,i (Yln,M (u, ·); Iln ej )
n,M,i n,M,i Φ du. − X(nr]/n ×Hln,M,j (X n,M ) Xl/n
252
Takashi Kato
Let r0 be such that (3.21)
1 1 1 = + . Since r0 p0 q 0
1 p0 − 3 − 2δ0 1 1 = > 0, 1+ − 2 p0 r0 2p0
using Proposition 2 with U = Fkn,i , V = Hln,M,i (X n,M ) and u = 1, we have (3.22)
n,(1) |Λk,l | ≤ C8 E n [ sup |Fkn,i (w)|p0 ]1/p0 + 1 |w|∞ ≤M
0 × E n [|Hln,M,i (X n,M )Φ|r0 ]1/r0 αk−l 0 ≤ C9 E n [Φq0 ]1/q0 αk−l .
for some C8 , C9 > 0 depending only on M . Also we see (3.23)
n,M,j n,M,i n,M,i n (X n,M )(Xl/n − X[nr]/n )Φ|r0 ]1/r0 E [|Hl n,M n,M,i n,M,i )Hln,j (X n,M )(Xl/n − X[nr]/n )Φ|r0 ]1/r0 = E n [|ϕM (Xl/n n,M ≤ M E n [|ϕM (Xl/n )Hln,j (X n,M )Φ|r0 ]1/r0
≤ M E n [|Hln,M,j (X n,M )|p0 ]1/p0 E n [Φq0 ]1/q0 . Then, using Proposition 2 again, we have (3.24)
n,(2)
n,(3)
0 |Λk,l |, |Λk,l | ≤ C10 E n [Φq0 ]1/q0 αk−l
for some C10 = C10 (M ) > 0. Thus (3.25)
J4n ≤ C11 ×
[nv] k−1 1 n q0 1/q0 0 αk−l E [Φ ] n k=[nr] l=[nr]
≤ 3C11
∞
αk0 |u − s| E n [Φq0 ]1/q0
k=1
for some C11 = C11 (M ) > 0. By the similar calculation of (3.23), we have (3.26)
J5n ≤ C12 |u − s| E n [Φq0 ]1/q0
Limits of Functional Stochastic Difference Equations
253
for some C12 = C12 (M ) > 0. Then the inequality (3.15) holds. Using (3.15) with v = u, r = t and Φ = |Xtn,M,i − n,M,i 1+δ0 Xs | 1{|X n,M |[nt]/n |≤M } , we get (3.27)
J0n ≤ C4 |u − s| E n [|Xtn,M,i − Xsn,M,i |p0 1{|X n,M
[nt]/n
]1/q0 .
|≤M }
Using (3.15) again with v = [nt]/n, r = s and Φ = 1, we get (3.28)
n,M,i n n,M,i 2 | ] ≤ C4 |u − s|. E [|X[nt]/n − Xs
Thus n,M,i n n,M,i p0 | 1{|X n,M |≤M } ] E [|Xt/n − Xs [nt]/n n,M,i ≤ C13 E n [|X[nt]/n − Xsn,M,i |p0 1{|X n,M |≤M } ] [nt]/n
+E
n
[|Xtn,M,i
−
n,M,i p0 X[nt]/n | 1{|X n,M |≤M } ] [nt]/n
n,M,i ≤ C14 M p0 −2 E n [|X[nt]/n − Xsn,M,i |2 ]
1 n,M,i (X n,M )|p0 ] + √ p (nt − [nt]) E n [|H[nt] ( n) 0 1 ≤ C15 |u − s| + √ p ≤ 2C15 |u − s| ( n) 0 for some C13 , C14 , C15 > 0 depending only on M . Thus the inequality (3.13) holds also when u − s ≥ 1/n. This completes the proof of Proposition 4. By Proposition 4, for any subsequence (nk )k , there is a further subsequence (nkl )l such that Qnkl ,M converges weakly to some probability measure QM on C([0, ∞); Rd ) as l → ∞ for each fixed M > 1 + |x0 |. d ) = 1. Proposition 5. QM (CM
Proof. For each T > 0, it follows that (3.29)
QM ( sup |w(t)| > M ) 0≤t≤T M
= lim Q ( sup |w(t)| > M + ε) ε0
0≤t≤T nkj
≤ lim lim inf P ε0 n→∞
( sup |Xtn,M | > M + ε). 0≤t≤T
254
Takashi Kato
Here we see P n ( sup |Xtn,M | > M + ε) 0≤t≤T
1 n,M n | ≤ M, |Xk/n | + √ |Hkn,M (X n,M )| > M + ε ≤ P n (|Xk/n n for some k = 0, . . . , [nT ])
[nT ]
≤
√ P n (|Hkn,M (X n,M )| ≥ ε n) ≤ C0 ×
k=0
1 √ ε3 n
for some C0 = C0 (M, T ) > 0. Thus QM ( sup |w(t)| > M ) = 0, T > 0.
(3.30)
0≤t≤T
This implies the assertion. Next we define functions aM,ij (t, w) and bM,i (t, w) by aM,ij (t, w) = ϕM (w(t))2 aij (t, w) b
M,i
(t, w) =
ϕM (w(t))bi0 (t, w)
d ϕM (w(t))2 B ij (t, w) + j=1
∂ ij +ϕM (w(t)) j ϕM (w(t))A (t, w) ∂x and let d d 1 M,ij ∂2 ∂ f (t, w) = a (t, w) i j f (w(t)) + bM,i (t, w) i f (w(t)) 2 ∂x ∂x ∂x M
i,j=1
i=1
for f ∈ C 2 (Rd ). Proposition 6. QM is a solution of the martingale problem associated with the generator M and starting at x0 . By Proposition 5, in order to prove Proposition 6, it suffices to show that (3.31)
QM
[(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))] t M = E Q [ M f (u, w)duΦ(w(s1 ), . . . , w(sN ))] E
s
Limits of Functional Stochastic Difference Equations
255
for any C ∞ function f : Rd −→ R with compact support, integer N , real numbers 0 ≤ s1 < . . . < sN ≤ s < t and bounded continuous function Φ : (RN )m −→ R. Until Proposition 14, we omit the M in Xtn,M and Ykn,M (u, t) as long as there is no misunderstanding, and simply denote (nkl ) by (n). Since f and Φ are bounded, it follows that (3.32)
E
Qn,M
[(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))] M
−→ E Q [(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))]. On the other hand, Taylor’s theorem implies Qn,M
[(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))] 1 1 1 = K1n + K2n + K3n + K4n + K5n + K6n + K7n + K8n , 2 2 2
(3.33)
E
where n K1n = E n [(f (Xtn ) − f (X[nt]/n ))Φ(Xsn1 , . . . , XsnN )], n ) − f (Xsn ))Φ(Xsn1 , . . . , XsnN )], K2n = E n [(f (X[ns]/n
K3n
d [nt]−1 1 n ∂ n,M,i n =√ (X n )Φ(Xsn1 , . . . , XsnN )], E [ i f (Xk/n )Fk ∂x n i=1 k=[ns]
K4n
d [nt]−1 1 n ∂ n,M,i n = (X n )Φ(Xsn1 , . . . , XsnN )], E [ i f (Xk/n )Gk n ∂x i=1 k=[ns]
K5n =
d [nt]−1 1 n ∂2 n E [ i j f (Xk/n ) n ∂x ∂x i,j=1 k=[ns]
× Fkn,M,i (X n )Fkn,M,j (X n )Φ(Xsn1 , . . . , XsnN )], K6n
d [nt]−1 1 n ∂2 n = √ E [ i j f (Xk/n ) ∂x ∂x n n i,j=1 k=[ns]
× Fkn,M,i (X n )Gn,M,j (X n )Φ(Xsn1 , . . . , XsnN )], k
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Takashi Kato
K7n
d [nt]−1 1 n ∂2 n = 2 E [ i j f (Xk/n ) n ∂x ∂x i,j=1 k=[ns]
(X n )Gn,M,j (X n )Φ(Xsn1 , . . . , XsnN )], × Gn,M,i k k [nt]−1 1 d 1 ∂3 n K8 = √ (1 − u)2 E n [ i j ν f (Ykn (u, k/n)) ∂x ∂x ∂x n n 0 i,j,ν=1 k=[ns]
× Hkn,M,i (X n )Hkn,M,j (X n )Hkn,M,ν (X n )Φ(Xsn1 , . . . , XsnN )]du.
Proposition 7. Kjn −→ 0 as n → ∞, j = 1, 2, 6, 7, 8. Proof. By (3.2) and (3.3), we have |K6n | ≤
1 √
n n
[nt]−1
C(M, f, Φ) −→ 0
k=[ns]
for some constant C(M, f, Φ) > 0. Similarly we get K7n −→ 0 and K8n −→ 0. Taylor’s theorem implies 1 √ n d
|K1n | ≤
i=1
0
1
∂
n,M,i n n f Y (u, t) (nt − [nt])H (X )Φ du E i [nt] [nt] ∂x n
1 ≤ const . × √ −→ 0. n Similar arguments give us K2n −→ 0. Then we obtain the assertion. To treat the convergent of K3n , K4n and K5n , we will show the following three propositions. Proposition 8. Let Ukn : C([0, ∞); Rd ) × Ωn −→ R be a continuously n Fr´echet differentiable random function such that Ukn (w) is Fk,∞ -measurable d n and E n [Ukn (w)] = 0 for each w ∈ CM , and V n : Ωn −→ R be an F0,[ns] measurable random variable. Suppose that there exists a constant C0 =
Limits of Functional Stochastic Difference Equations
C0 (M ) > 0 such that sup εγ Nn (ε, M ; Ukn ) ≤ C0 ,
(3.34)
ε>0
sup sup εγ Nn (ε, M ; ∇Ukn (·; Iln ej )) ≤ C0 , l≤k ε>0
1
(3.35)
E
m=0
n
sup |∇m Ukn (w)|pL0m
|w|∞ ≤M
k/n
≤ C0
and n n p /2 E [|V | 0 ] ≤ C0
(3.36)
for any j = 1, . . . , d, n ∈ N and k ∈ Z+ . Then it holds that [nt]−1 1 n β n n n n E [D ϕM (Xk/n )Uk (X )V ] −→ 0, n → ∞ n
(3.37)
k=[ns]
for β ∈ Zd+ with |β| ≤ 1. Proof. By Taylor’s theorem, we have n β n n n n E [D ϕM (Xk/n )Uk (X )V ]
=
k−1
n β n n n E [{D ϕM (X(l+1)/n )Uk (X·∧((l+1)/n) )
l=[ns] n n )Ukn (X·∧(l/n) )}V n ] −Dβ ϕM (Xl/n n n )Ukn (X·∧([ns]/n) )V n ] + E n [Dβ ϕM (X[ns]/n d k−1 1 1 n ∂ β n,M (u, k/n)) = √ E [ i D ϕM (Yl ∂x n 0 i=1 l=[ns]
×Ukn (Yln,M (u, ·))Hln,M,i (X n )V n ] + E n [Dβ ϕM (Yln,M (u, k/n))
×∇Ukn (Yln,M (u, ·); Iln ei )Hln,M,i (X n )V n ] du
n n + E n [Dβ ϕM (X[ns]/n )Ukn (X·∧([ns]/n) )V n ].
257
258
Takashi Kato
By Proposition 2, we see that ∂ (3.38) E n [ i Dβ ϕM (Yln,M (u, k/n))Ukn (Yln,M (u, ·))Hln,M,i (X n )V n ] ∂x 0 ≤ C1 αk−l , (3.39) E n [Dβ ϕM (Yln,M (u, k/n))∇Ukn (Yln,M (u, ·); Iln ei )Hln,M,i (X n )V n ] 0 ≤ C1 αk−l and (3.40)
n β E [D ϕM (X n
n n n [ns]/n )Uk (X·∧([ns])/n )V ]
0 ≤ C1 αk−[ns]
for some C1 > 0 depending only on M and C0 . Thus [nt]−1 1 n β n n n n E [D ϕM (Xk/n )Uk (X )V ] n k=[ns]
[nt]−1 k−1 1 1 0 0 √ ≤ 2C1 d × + αk−[ns] α n n k−l k=[ns]
≤ 2C1 d
∞ k=1
l=[ns]
1 αk0 (t + 1) × √ −→ 0, n → ∞. n
Then we obtain the assertion. Proposition 9. Let Ukn , Vkn : C([0, ∞); Rd ) × Ωn −→ R be such that n Ukn (w) and Vkn (w) are Fk,k -measurable and continuously Fr´echet differend tiable random functions such that E n [Ukn (w)] = 0 for each w ∈ CM , and n Z n : Ωn −→ R be an F0,[ns] -measurable random variable. Suppose that there exists a constant C0 = C0 (M ) > 0 such that sup εγ Nn (ε, M ; Ukn ) + Nn (ε, M ; Vkn ) ≤ C0 , (3.41) ε>0 sup sup εγ Nn (ε, M ; ∇Ukn (·; Iln ej )) (3.42) l≤k ε>0 +Nn (ε, M ; ∇Vkn (·; Iln ej )) ≤ C0 , 1 n (3.43) sup |∇m Ukn (w)|pL0m E |w|∞ ≤M
m=0
≤ C0 ,
1 m=0
E
n
k/n
sup |∇m Vkn (w)|pL0m
|w|∞ ≤M
k/n
≤ C0
Limits of Functional Stochastic Difference Equations
259
and n n p E [|Z | 0 ] ≤ C0
(3.44)
for any j = 1, . . . , d, n ∈ N and k ∈ Z+ . Then it holds that (3.45)
(i)
[nt]−1 1 n β n β n n n n E [D ϕM (Xk/n )D ϕM (Xk/n )Ξkk (X )Z ] −→ 0, n k=[ns]
(3.46)
(ii)
[nt]−1 k−1 1 n β n E [D ϕM (Xl/n ) n k=[ns] l=[ns]
n n )Ξnkl (X·∧(l/n) )Z n ] −→ 0 ×Dβ ϕM (Xl/n
as n → ∞ for β, β ∈ Zd+ with |β| + |β | ≤ 1, where Ξnkl (w) = Ukn (w)Vln (w) − n n n E [Uk (w)Vl (w)]. Proof. By Taylor’s theorem, we have
n β n β n n n n E [D ϕM (Xl/n )D ϕM (Xl/n )Ξkl (X·∧(l/n) )Z ]
=
l−1
n β n β n n n E [{D ϕM (X(m+1)/n )D ϕM (X(m+1)/n )Ξkl (X·∧((m+1)/n) )
m=[ns]
n n n )Dβ ϕM (Xm/n )Ξnkl (X·∧(m/n) )}Z n ] −Dβ ϕM (Xm/n
n n n )Dβ ϕM (X[ns]/n )Ξnkl (X·∧([ns])/n )Z n ] + E n [Dβ ϕM (X[ns]/n d l−1 1 ∂ 1 n D β ϕM D β ϕM = √ E i ∂x n i=1 m=[ns] 0 ∂ +Dβ ϕM i Dβ ϕM (Ymn,M (u, l/n)) ∂x n n,M n,M,i (X n )Z n ×Ξkl (Ym (u, ·))Hm + E n Dβ ϕM (Ymn,M (u, l/n))Dβ ϕM (Ymn,M (u, l/n)) n n,M,i ei )Hm (X n )Z n du ×∇Ξnkl (Ymn,M (u, ·); Im
n n n )Dβ ϕM (X[ns]/n )Ξnkl (X·∧([ns])/n )Z n ]. + E n [Dβ ϕM (X[ns]/n
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Takashi Kato
Since n ∇Ξnkl (w; Im ei )
(3.47)
n n = ∇Ukn (w; Im ei )Vln (w) − E n [∇Ukn (w; Im ei )Vln (w)] n n +Ukn (w)∇Vln (w; Im ei ) − E n [Ukn (w)∇Vln (w; Im ei )]
holds, using Proposition 3, we get n β n E [D ϕM (X n )Dβ ϕM (X n )Ξn (X n (3.48) )Z ] kl l/n l/n ·∧(l/n) l−1 1 0 0 0 0 αk−l αl−m + αk−l αl−[ns] ≤ C1 √ n m=[ns]
for some C1 > 0 depending only on M and C0 . In particular it follows that n β E [D ϕM (X n )Dβ ϕM (X n )Ξn (X n )Z n ] (3.49) kk k/n k/n k−1 1 0 0 . αk−m + αk−[ns] ≤ C1 √ n m=[ns]
Thus we have [nt]−1 1 n β n β n n n n E [D ϕM (Xk/n )D ϕM (Xk/n )Ξkk (X )Z ] n k=[ns] ∞
≤ 2C1
k=1
1 αk0 (t + 1) × √ −→ 0, n → ∞ n
and [nt]−1 k−1 1 n β n β n n n n E [D ϕM (Xl/n )D ϕM (Xl/n )Ξkl (X·∧(l/n) )Z ] n
≤
k=[ns] l=[ns] ∞
2 2C1 αk0 (t k=1
1 + 1) × √ −→ 0, n → ∞. n
Then we obtain the assertion. Proposition 10. Let ψ : Rd −→ R be a continuously differentiable function such that ψ(x) = 0 for any x ∈ Rd with |x| > M and g n :
Limits of Functional Stochastic Difference Equations
261
Z+ × C([0, ∞); Rd ) −→ R, g : [0, ∞) × C([0, ∞); Rd ) −→ R be functionals. Suppose that g n (k, ·) is Bk/n -measurable and continuous, and that there exists a constant C0 = C0 (M ) > 0 such that sup |g n (k, w)| ≤ C0
(3.50)
|w|∞ ≤M
for each n ∈ N and k ∈ Z+ . Moreover suppose sup |g n ([nt], w) − g(t, w)| −→ 0, n → ∞
(3.51)
w∈K
for each K ∈ Kd and t ≥ 0. Then it holds that [nt]−1 1 n n n n n n E [ψ(Xk/n )g (k, X )Φ(Xs1 , . . . , XsN )] n k=[ns] t QM [ψ(w(u))g(u, w)Φ(w(s1 ), . . . , w(sN ))]du, n → ∞ −→ E
(3.52)
s
Proof. Denote the left-hand side of (3.52) by K n . Define Ln and S n by n
L = s
and
t
n n n n n n E [ψ(Xk/n )g ([nu], X )Φ(Xs1 , . . . , XsN )]du
n
S = s
t
n n n n n E [ψ(Xu )g(u, X )Φ(Xs1 , . . . , XsN )]du.
Then we have
|K − L | ≤ C0 n
n
s
t
n n n E [|ψ(Xu ) − ψ(X[nu]/n )| · |Φ|]du
1 ≤ const . × √ n d
i=1
t
∂ n (v, u)) i ψ(Y[nu] ∂x s 0 n,M,j (X n ) dvdu ×(nu − [nu])H[nu]
1 ≤ const . × √ −→ 0. n
1
E
n
262
Takashi Kato
Next we will show Ln − S n −→ 0.
(3.53)
Take any ε > 0. Then, by Proposition 4, there exists a compact set K ⊂ C([0, ∞); Rd ) such that inf Qn,M (K) > 1 − ε.
(3.54)
n
d Set KM = K ∩ CM . Then, by Proposition 1, we have n E [ψ(Xun )(g n ([nu], X n ) − g(u, X n ))Φ] ≤ const . × sup |g n ([nu], w) − g(u, w)| w∈KM + E n [ψ(Xun )(g n ([nu], X n ) − g(u, X n )); X n ∈ / K] ≤ const . × sup |g n ([nu], w) − g(u, w)| w∈KM n + sup |g ([nu], w)| + |g(u, w)| ε . |w|∞ ≤M
for each u ∈ [s, t]. Since KM ∈ Kd holds, by (3.50), we have (3.55) lim sup E n [ψ(Xun )(g n ([nu], X n ) − g(u, X n ))Φ] ≤ const . × ε. n→∞
Thus (3.56)
lim E n [ψ(Xun )(g n ([nu], X n ) − g(u, X n ))Φ] = 0
n→∞
for each u ∈ [s, t]. By (3.50) again and the bounded convergence theorem, we get (3.57)
|Ln − S n | t n E [ψ(Xun )(g n ([nu], X n ) − g(u, X n ))Φ]du −→ 0. ≤ s
Since
t
ψ(w(u))g(u, w)Φ(w(s1 ), . . . , w(sN ))du
F (w) = s
Limits of Functional Stochastic Difference Equations
is continuous and Proposition 1 implies Qn,M (|F (w)| ≤ C1 ) = 1
(3.58) for each n ∈ N, where
C1 = C0 |t − s| sup |ψ(x)| |x|≤M
sup
y1 ,... ,yN ∈Rd
|Φ(y1 , . . . , yN )|,
using the continuous mapping theorem, we get t n QM S −→ [ψ(w(u))g(u, w)Φ(w(s1 ), . . . , w(sN ))]du. E s
This completes the proof of Proposition 10. By Proposition 8, 9(i) and 10, we have the following. Proposition 11. d t ∂ n QM f (w(u))ϕM (w(u)) (i) K4 −→ E ∂xi s
×bi0 (u, w)Φ(w(s1 ), . . . , w(sN )) du,
i=1
(ii) K5n −→
d
t
E
QM
i,j=1 s
as n → ∞.
∂2 f (w(u))ϕM (w(u))2 ∂xi ∂xj
×aij (u, w)Φ(w(s ), . . . , w(s )) du 1 N 0
Next we calculate the limit of K3n . Using Taylor’s theorem, we have n n n n n n n n K3n = K3,1 + K3,2 + K3,3 + K3,4 + K3,5 + K3,6 + K3,7 + K3,8 ,
where n K3,1
d [nt]−1 1 n ∂ n,i n n n =√ E [ i f (X[ns]/n )ϕM (X[ns]/n )Fk (X·∧([ns]/n) )Φ], ∂x n i=1 k=[ns]
n K3,2
d [nt]−1 k−1 1 n ∂2 n n 2 = E [ i j f (Xl/n )ϕM (Xl/n ) n ∂x ∂x i,j=1 k=[ns] l=[ns]
n × Fkn,i (X·∧(l/n) )Fln,j (X n )Φ],
263
264
n K3,3
Takashi Kato d [nt]−1 k−1 1 n ∂2 n n 2 √ = E [ i j f (Xl/n )ϕM (Xl/n ) ∂x ∂x n n i,j=1 k=[ns] l=[ns]
n n × Fkn,i (X·∧(l/n) )Gn,j l (X )Φ], n K3,4 =
d [nt]−1 k−1 1 n ∂ n n E [ i f (Xl/n )ϕM (Xl/n ) n ∂x i,j=1 k=[ns] l=[ns]
× n K3,5
∂ n n ϕM (Xl/n )Fkn,i (X·∧(l/n) )Fln,j (X n )Φ], ∂xj
d [nt]−1 k−1 1 n ∂ n n = √ E [ i f (Xl/n )ϕM (Xl/n ) ∂x n n i,j=1 k=[ns] l=[ns]
× n K3,6
∂ n n n ϕM (Xl/n )Fkn,i (X·∧(l/n) )Gn,j l (X )Φ], j ∂x
d [nt]−1 k−1 1 n ∂ n n 2 = E [ i f (Xl/n )ϕM (Xl/n ) n ∂x i,j=1 k=[ns] l=[ns]
n × ∇Fkn,i (X·∧(l/n) ; Iln ej )Fln,j (X n )Φ], n K3,7
d [nt]−1 k−1 1 n ∂ n n 2 = √ E [ i f (Xl/n )ϕM (Xl/n ) ∂x n n i,j=1 k=[ns] l=[ns]
n K3,8
d 1 = √ n n
[nt]−1 k−1
i,j,ν=1 k=[ns] l=[ns] 0
n n × ∇Fkn,i (X·∧(l/n) ; Iln ej )Gn,j l (X )Φ], 1
n,M,ijν (1 − u) E n [ηkl (Yln (u, ·))
× Hln,M,j (X n )Hln,M,ν (X n )Φ]du and n,M,ijν (w) = ηkl
∂3 f (w(l/n))Fkn,M,i (w) ∂xi ∂xj ∂xν ∂2 + i j f (w(l/n))∇Fkn,M,i (w; Iln eν ) ∂x ∂x ∂2 + i ν f (w(l/n))∇Fkn,M,i (w; Iln ej ) ∂x ∂x ∂ + i f (w(l/n))∇2 Fkn,M,i (w; Iln ej , Iln eν ). ∂x
Limits of Functional Stochastic Difference Equations
265
n Proposition 12. K3,j −→ 0 as n → ∞, j = 1, 3, 5, 7, 8.
Proof. Applying Proposition 2 with U
Fkn,i and V
=
∂ n f (X[ns]/n )Φ, we have ∂xi n | |K3,1
[nt]−1 ∞
1 1 0 ≤ const . × √ αk−[ns] ≤ const . × αk0 √ −→ 0. n n k=0
k=[ns]
Applying Proposition 2 again with U ∂2 n n n f (Xl/n )ϕM (Xl/n )Gn,j l (X )Φ, we have ∂xi ∂xj n | |K3,3
=
Fkn,i
=
and V
=
[nt]−1 k−1 ∞
1 1 0 ≤ const . × √ αk−l ≤ const . × αk0 √ −→ 0. n n n k=0
k=[ns] l=[ns]
n,M,ijν n n Similarly we have K3,5 −→ 0 and K3,7 −→ 0. Since ηkl (w) is the finite sum of the following terms
Dβ f (w(l/n))Dβ ϕM (w(k/n))U (w) with β, β ∈ Zd+ and U (w) = Fkn,i (w), ∇Fkn,i (w; Iln ej ) or ∇2 Fkn,i (w; Iln ej , n Iln eν ), by Proposition 2, it follows that K3,8 −→ 0. Then we obtain the assertion. n n n For K3,2 , K3,4 and K3,6 , we will show the following proposition.
Proposition 13. Let ψ : Rd −→ R be a continuously differentiable n function such that ψ(x) = 0 for any x ∈ Rd with |x| > M , and ξk,l : C([0, ∞); Rd ) −→ R, k, l ∈ Z+ , Ξ : [0, ∞) × C([0, ∞); Rd ) −→ R be funcn is Bl/n -measurable and continuous, and that there tionals. Suppose that ξk,l exists a constant C0 = C0 (M ) > 0 such that (3.59)
∞
sup
n sup |ξk,l (w)| ≤ C0
k=1 l∈Z+ |w|∞ ≤M
266
Takashi Kato
for each n ∈ N. Moreover suppose ∞ n ξk,[nt] (w) − Ξ(t, w) −→ 0, n → ∞ sup
(3.60)
w∈K
k=1
for each K ∈ Kd and t ≥ 0. Then it holds that [nt]−1 k−1 1 n n n n n n (3.61) E [ψ(Xl/n )ξk−l,l (X )Φ(Xs1 , . . . , XsN )] n k=[ns] l=[ns] t QM [ψ(w(u))Ξ(u, w)Φ(w(s1 ), . . . , w(sN ))]du, n → ∞. −→ E s
Proof. Denote the left-hand side of (3.61) by U n and set V
n
[nt]−1 ∞ 1 n n n n n n = E [ψ(Xl/n )ξk,l (X )Φ(Xs1 , . . . , XsN )]. n l=[ns] k=1
Since Fubini’s theorem implies (3.62)
[nt]−2 [nt]−l−1 1 n n n n n n U = E [ψ(Xl/n )ξk,l (X )Φ(Xs1 , . . . , XsN )], n n
l=[ns]
k=1
we have (3.63)
|U n − V n | ≤ C1 (M, ψ, Φ)
1 n
+
t
∞
sup
n sup |ξk,l (w)|du
s k=[nt]−[nu] l∈Z+ |w|∞ ≤M
for some C1 (M, ψ, Φ) > 0. By (3.59), the integrand in the right-hand side of (3.63) is bounded and converges to zero as n → ∞ for u ∈ [s, t). Thus, using the bounded convergence theorem, we have U n − V n −→ 0.
(3.64)
Since Proposition 10 implies t QM (3.65) V n −→ [ψ(w(u))Ξ(u, w)Φ(w(s1 ), . . . , w(sN ))]du, E s
Limits of Functional Stochastic Difference Equations
267
we have our assertion. Proposition 14. d t ∂2 n QM [ i j f (w(u))ϕM (w(u))2 (i) K3,2 −→ E ∂x x s i,j=1
n (ii) K3,4 −→
(iii)
n K3,6
−→
d
×Aij (u, w)Φ(w(s1 ), . . . , w(sN ))]du, t
E
QM
[
i,j=1 s d
∂ ∂ f (w(u))ϕM (w(u)) j ϕM (w(u)) i ∂x ∂x ×Aij (u, w)Φ(w(s1 ), . . . , w(sN ))]du,
t
E
QM
i,j=1 s
[
∂ f (w(u))ϕM (w(u))2 ϕM (w(u)) i ∂x ×B ij (u, w)Φ(w(s1 ), . . . , w(sN ))]du
as n → ∞. n,ij Proof. Define ξk,l by n,ij ξk,l
=E
n
n,i Fk+l
l
n,j Fl (w) . w ·∧ n
By assumption [A7], we have (3.66)
∞ n,ij n,ij sup ξk,[nt] (w) − A (t, w) −→ 0, n → ∞
w∈K
k=1
for any K ∈ Kd and t ≥ 0. By Proposition 9, it follows that n n − K3,2,1 −→ 0, n → ∞ K3,2
(3.67) where n K3,2,1 =
d [nt]−1 k−1 1 n ∂2 n n 2 E [ i j f (Xl/n )ϕM (Xl/n ) n ∂x x i,j=1 k=[ns] l=[ns]
n,ij ×ξk−l,l (X n )Φ(Xsn1 , . . . , XsnN )].
Since Lemma 1 implies 1/3
n,ij n,i (w)| ≤ 8 E n [|Fk+l (w)|3 ]1/3 E n [|Fln,j (w)|3 ]1/3 αk , |ξk,l
268
Takashi Kato
we have ∞
(3.68)
sup
sup
k=1 l∈Z+ |w|∞ ≤M
n,ij |ξk,l (w)|
≤ C0
∞
1/3
αk
k=1
for some C0 = C0 (M ) > 0. Then, applying Proposition 13, we get (3.69)
n K3,2,1
−→
d
t
E
QM
i,j=1 s
[
∂2 f (w(u))ϕM (w(u))2 ∂xi xj ×Aij (u, w)Φ(w(s1 ), . . . , w(sN ))]du.
Then we obtain the assertion (i). The assertions (ii) and (iii) follow by the same way. By Proposition 7, 11, 12 and 14, it follows that (3.70)
Qn,M
[(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))] t QM −→ E [ M f (u, w)duΦ(w(s1 ), . . . , w(sN ))]. E
s
The equality (3.31) now follows by (3.32) and (3.70). This completes the proof of Proposition 6. Proposition 15. The family of measures (QM )M >1+|x0 | is tight on C([0, ∞); Rd ). Proof. We define the matrix σ M (t, w) = (σ M,ij (t, w))di,j=1 by σ M (t, w) = ϕM (w(t))a1/2 (t, w), where a1/2 (t, w) is the square root matrix of a(t, w). By Proposition 6, there exists the weak solution (ΩM , F M , (FtM )t , P M , (BtM )t , (XtM )t ) of the following stochastic differential equation dXtM = σ M (t, X M )dBtM + bM (t, X M )dt (3.71) X0M = x0 such that the distribution of X M under P M is equal to QM . Let T > 0. We will show that there exists a constant C0 (T ) > 0 such that (3.72)
E
M
[ sup |XtM |4 ] ≤ C0 (T ) 0≤t≤T
Limits of Functional Stochastic Difference Equations
269
Fix any R > 0 and define the stopping time τR and the function mR (t) by τR = inf{t ∈ R+ ; |XtM | ≥ R}. and M mR (t) = E M [ sup |Xs∧τ |4 ], R 0≤s≤t
M
where E denotes the expectation under P M . By the continuity of X M , we see that τR −→ ∞ as R → ∞ almost surely under P M . By the assumption [A8], the H¨ older inequality and the Burkholder-Davis-Gundy inequality, we have s∧τR 4 M mR (t) ≤ C1 E sup σ M (u, X M )dBuM +EM
0≤s≤t
sup
0≤s≤t
0
0 s∧τR
4 bM (u, X M )du
t M 1{s≤τR } |σ M (s, X M )|4 ds ≤ C1 t E 0 t +t3 E M 1{s≤τR } |bM (s, X M )|4 ds 0 t 1{s≤τR } (1 + sup |XuM |)4 ds ≤ C2 (T ) E M 0
t ≤ C3 (T ) 1 + mR (s)ds
0≤u≤s
0
for each t ≤ T and for some constants C1 , C2 (T ), C3 (T ) > 0. Applying the Gronwall inequality, we see sup mR (t) ≤ C4 (T )
(3.73)
0≤t≤T
for some C4 (T ) > 0. Letting R → ∞, we get (3.72) by Fatou’s lemma. Then, using the H¨ older inequality and the Burkholder-Davis-Gundy inequality again, we have PM
[|XtM − XsM |4 ] 4 t M ≤ C1 E 1{u≥s} σ M (u, X M )dBuM 0 4 t bM (u, X M )du + EM E
s
270
Takashi Kato
t ≤ C1 |t − s| E M [ |σ M (u, X M )|4 du] s t 3 M + |t − s| E [ |bM (u, X M )|4 du] s t
1 + E M [ sup |XvM |4 ] du ≤ C0 (T )C5 (T )|t − s|2 ≤ C5 (T )|t − s| 0≤v≤u
s
for some C5 (T ) > 0. Obviously QM (w ∈ C([0, ∞); Rd ); w(0) = x0 ) = 1 holds for all M . Then, using theorem 2.3 in [13], we obtain the tightness of (QM )M >1+|x0 | . Proof of Theorem 1. Proposition 15 implies that for any subsequence (Mk )k , there exists a further subsequence (Mkl )l such that QMkl converges to some probability measure Q∗ on C([0, ∞); Rd ). Take M0 large enough so that the support of f is contained in {x ∈ d R ; |x| ≤ M0 /2}. Since M f = f holds for M > M0 , by (3.31), it follows that (3.74)
E = E
Mk l
Q
M Q kl
[(f (w(t)) − f (w(s)))Φ(w(s1 ), . . . , w(sN ))] t [ f (u, w)duΦ(w(s1 ), . . . , w(sN ))] s
for Mkl > M0 . Letting l → ∞, we see that Q∗ is a solution of the martingale problem associated with the generator . Moreover, by the assumption [A10], Q∗ equals to Q and is independent of a subsequence (Mkl )l . Then it follows that QM converges weakly to Q on C([0, ∞); Rd ) as M → ∞. Finally, repeating the arguments in [5] p.119-120, we show that Qn converges weakly to Q on C([0, ∞); Rd ). This completes the proof of Theorem 1. 4.
Proof of Theorem 2
To prove Theorem 2, we will show two lemmas below. Let (Ω, F, P ) be a probability space and (S, d) be a metric space. 1 1 + < 1 and U : S × Ω −→ p q R be a continuous random function such that U (x) is A-measurable and Lemma 5. Let 1 < p, q < ∞ be such that
Limits of Functional Stochastic Difference Equations
271
E[U (x)] = 0 for each x ∈ S, and X : Ω −→ S, V : Ω −→ R be B-measurable random variables. Suppose that there exist positive constants C0 and γ such that sup εγ log N (ε, p; U ) ≤ C0 .
(4.1)
ε>0
Then for each ( ∈ (0, 1/γ) there exists a constant C > 0 depending only on p, q, γ, ( and C0 such that (4.2)
E[U (X)V ]
≤ C E[sup |U (x)|p ]1/p + 1 E[|V |q ]1/q x∈S
1 . log(1/α(A, B))
Proof. We may assume that the right-hand side of (4.2) is finite. Set 1 ξ= . Using Lemma 2 with ε = ξ , we have log(1/α(A, B))
(4.3) E[U (X)V ] ≤ 8 E[sup |U (x)|p ]1/p + 1 x∈S
× E[|V |q ]1/q ξ + ξ (1−r) exp(C0 ξ −γ − ξ −1 ) , 1 1 1 = 1 − − . Since (γ ∈ (0, 1) and ξ ∈ (0, 1), there is a constant r p q C1 > 0 which depends only on p, q, γ, ( and C0 such that
where
ξ (1−r) exp(C0 ξ −γ − ξ −1 ) ≤ C1 ξ .
(4.4)
By (4.3) and (4.4), we obtain our assertion. 1 1 1 + + < 1. Let p q r U, V : S × Ω −→ R be continuous random functions such that U (x) and V (x) are A and B-measurable respectively and E[U (x)] = 0 for each x ∈ S, and X : Ω −→ S, Z : Ω −→ R be C-measurable random variables. Suppose that there exist positive constants C0 , u∗ , v ∗ > 0 and γ such that Lemma 6. Let 1 < p, q, r < ∞ be such that
(4.5)
sup εγ log N (ε, p; U ) + log N (ε, q; V ) ≤ C0 , ε>0
272
Takashi Kato p 1/p ∗ E[sup |U (x)| ] ≤ u
(4.6)
x∈S
and q 1/q ∗ E[sup |V (x)| ] ≤ v .
(4.7)
x∈S
1 there exists a constant C > 0 depending only 2γ on p, q, r, γ, ( , u∗ , v ∗ and C0 such that E[Ξ(X)Z] (4.8)
1 1 ≤ C E[|Z|r ]1/r , log(1/α(A ∨ B, C)) log(1/α(A, B ∨ C)) Then for each ( ∈ 0,
where Ξ(x) = U (x)V (x) − E[U (x)V (x)]. Proof. By (2.17), we have (4.9)
sup εγ log N (ε, p; Ξ) ≤ 2γ+1 C0 (u∗ + v ∗ )γ . ε>0
Then, by Lemma 5, we see that (4.10)
E[Ξ(X)Z] ≤ C1 E[|Z|r ]1/r
2 1 log(1/α(A ∨ B, C))
for some C1 = C1 (p, q, r, γ, ( , u∗ , v ∗ , C0 ) > 0. By Lemma 1 and Lemma 5, we have (4.11) E[Ξ(X)Z]
2 1 ≤ C2 E[|Z|r ]1/r α(A, B ∨ C)1−1/p−1/q + log(1/α(A, B ∨ C)) for some C2 = C2 (p, q, r, γ, ( , u∗ , v ∗ , C0 ) > 0. C3 (p, q, ( ) > 0 such that
2 1 t1−1/p−1/q ≤ C3 (4.12) log(1/t)
Since there is C3 =
for all t ∈ (0, 1/4], we get (4.13)
E[Ξ(X)Z] ≤ C2 (C3 + 1) E[|Z|r ]1/r
2 1 . log(1/α(A, B ∨ C))
Limits of Functional Stochastic Difference Equations
273
By (4.10) and (4.13), we obtain the assertion. By Lemma 5, Lemma 6 and the same arguments in the proof of Theorem 1, we obtain Theorem 2. 5.
Appendix
5.1. Sufficient conditions for [A9] Let a(t, w) = (aij (t, w))dij=1 and b(t, w) = (bi (t, w))di=1 be as in [A8], and let σ(t, w) = (σ ij (t, w))di,j=1 = a1/2 (t, w). It is well-known that if we assume the Lipschitz condition of σ ij (t, w) and bi (t, w), then the condition [A9] holds. In fact, the local Lipschitz continuity of bi (t, w) is obtained by [A3] and [A5]. In this section we introduce the sufficient condition under which σ ij (t, w) is Lipschitz continuous. [A10] aij (t, w) is twice continuously Fr´echet differentiable in w for each t ≥ 0, and for each T > 0 there exists a positive constant C(T ) > 0 such that |∇2w aij (t, w)|L2t ≤ C(T )
(5.1)
for each t ∈ [0, T ] and w ∈ C([0, ∞); Rd ), where ∇2w aij (t, w) denotes the second Fr´echet derivative of aij (t, w) with respect to w. Here we remark that since aij (t, ·) is measurable with respect to Bt , we can regard ∇2w aij (t, w) as the element of L2t for each fixed t ≥ 0. Theorem 3. Assume [A1] − [A8] and [A10]. Then the conclusion of Theorem 1 holds. Proof. Let σ(t, w) = a1/2 (t, w). To check the condition [A9], it suffices to show that for each M > 0 and T > 0 there exists a constant C0 = C0 (M, T ) > 0 such that (5.2)
|σ ij (t, w) − σ ij (t, w )| ≤ C0 sup |w(s) − w (s)|,
(5.3)
|b (t, w) − b (t, w )| ≤ C0 sup |w(s) − w (s)|
0≤s≤t
i
i
0≤s≤t
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Takashi Kato
d for any t ∈ [0, T ] and w, w ∈ CM . By [A3], we have
(5.4)
|∇w bn,i 0 (k, w)|L1
k/n
d ≤ E n [|∇Gn,i k (w)|L1 ] ≤ C1 , k ∈ Z+ , w ∈ CM k/n
for some C1 = C1 (M ) > 0. Moreover, by [A3], [A5] and Lemma 1, we have (5.5) |∇w B n,ij (k, w)|L1 ≤
k/n
k
3 1/3 n n,j n 2 n,i 3 1/3 2 E ∇ Fk+l w · ∧ E [|Fk (w)| ] n Lk/n k
3 1/3 n 1/3 n,i n,j n 3 1/3 αl + E ∇Fk+l w · ∧ 1 E [|∇Fk (w)|L1 ] k/n n Lk/n
∞ l=1
≤ C2
∞
1/3
d αl , k ∈ Z+ , w ∈ CM
l=1
for some C2 = C2 (M ) > 0. By (5.4) and (5.5), we get (5.3). To see (5.2), we introduce the following theorem (Theorem 5.2.3 in [14]). Theorem 4. Let f (t, x) = (f ij (t, x))di,j=1 : [0, T ] × R −→ Rd ⊗ Rd be a symmetric non-negative definite matrix-valued function. Suppose that f ij (t, x) is twice continuously differentiable in x for each t ≥ 0 and that there is a positive constant C(T ) such that ∂2 (5.6) 2 f ij (t, x) ≤ C(T ) ∂x for each t ∈ [0, T ], x ∈ R and i, j = 1, . . . , d. Then it holds that |g ij (t, x) − g ij (t, y)| ≤ d 2C(T )|x − y| (5.7) for each t ∈ [0, T ] and x, y ∈ R, where g(t, x) = f 1/2 (t, x). For each fixed T > 0 and w, w ∈ C([0, ∞); Rd ), define the functions f, g : [0, T ] × R −→ Rd ⊗ Rd by f (t, x) = a(t, w + x(w − w )) and g(t, x) = f 1/2 (t, x). By [A10], f (t, x) is twice continuously differentiable in x for each t and d2 ij (5.8) 2 f (t, x) = |∇2w aij (t, w + x(w − w ); w − w , w − w )| dx ≤ C4 sup |w(s) − w (s)|2 , t ∈ [0, T ], x ∈ R 0≤s≤t
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for some C4 (T ) > 0. Then Theorem 4 implies |σ ij (t, w) − σ ij (t, w )| = |g ij (t, 1) − g ij (t, 0)| ≤ d
2C4 sup |w(s) − w (s)|. 0≤s≤t
This implies (5.2). Then the condition [A9] holds and we obtain the conclusion. 5.2. Sufficient conditions for [A4] and [B4] In this section we provide sufficient conditions under which [A4] and [B4] are filled. Let ε > 0, (S, d) be a metric space and A be a totally bounded subset m of S. We say that a family of sets (Ai )m is an ε-net of A if A ⊂ Ai i=1 i=1
ˆ (ε; A, d) the and sup d(x, y) < ε for each i = 1, . . . , m. We denote by N x,y∈Ai
minimum of cardinals of ε-nets of A in the metric d. Theorem 5. Let (Ω, F, P ) be a probability space, p ≥ 1, (S, d) be a metric space, (B, ||·||B ) be a Banach space and A be a totally bounded subset of B. Let f : B × Ω −→ R be a continuously Fr´echet differentiable random function and u : S −→ B be a continuous function such that u(x) ∈ A for any x ∈ S. Suppose that there exists a positive constant C0 such that p 1/p E[sup ||∇f (y)||B ∗ ] ≤ C0 ,
(5.9)
˜ y∈A
where A˜ is a convex hull of A and ||∇f (y)||B ∗ =
|∇f (y; z)| , y ∈ B. ||z||B z∈B,z=0 sup
Then for any ε > 0 (5.10)
ˆ (ε/C0 ; A, dB ), N (ε, p; U ) ≤ N
where U (x, ω) = f (u(x), ω) and dB (y, y ) = ||y − y ||B , y, y ∈ B. Proof. Let (Ai )m i=1 be an ε-net of A. We define Si ⊂ S by Si = {x ∈ S ; u(x) ∈ Ai }.
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Takashi Kato
Then we have S=
(5.11)
m
Si
i=1
and for each x, x ∈ Si |U (x) − U (x )| ≤
1
||∇f (tu(x) + (1 − t)u(x ))||B ∗ dt||u(x) − u(x )||B
0
≤ sup ||∇f (y)||B ∗ × ε. ˜ y∈A
Then we have (5.12)
E[ max
sup |U (x) − U (x )|p ]1/p ≤ C0 ε.
i=1,... ,m x,x ∈Si
By (5.11) and (5.12), we see that (Si )m i=1 is an (C0 ε, p, U )-net of S. Then we obtain the assertion. Let B be a Banach space and B(B) be a Borel field of B. By Theorem 5, under suitable conditions, we can check conditions [A4] and [B4] when Fkn,i and Gn,i k are represented in the following form n,i (5.13) Fkn,i (w, ω) = fkn,i (u(k/n, w), ω), Gn,i k (w, ω) = gk (v(k/n, w), ω),
where fkn,i (x, ω), gkn,i (x, ω) : B ×Ωn −→ R be B(B)⊗F n -measurable random functions and u(t, w), v(t, w) : [0, ∞)×C([0, ∞); Rd ) −→ B be (Bt )t -adapted (i.e. u(t, ·) and v(t, ·) are Bt -measurable for each t ≥ 0) deterministic functions. We also have the condition [A4] when the image spaces of Fkn,i and Gn,i k are finite dimensional in Lp0 (Ωn ). Let p ≥ 1, (Ω, F, P ) be a probability space, (S, d) be a metric space and U : S × Ω −→ R be a continuous random function which satisfies E[|U (x)|p ] < ∞ for any x ∈ S. We define the metric space (Sp (U ), dp ) by Sp (U ) = {U (x) ∈ Lp (Ω) ; x ∈ S} and dp (X, Y ) = E[|X − Y |p ]1/p .
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Theorem 6. Suppose that there are constants γ ∈ (0, p/2), C0 > 0 and C1 > 0 such that ˆ (ε; Sp (U ), dp ) ≤ C0 sup εγ N
(5.14)
ε>0
and p E[sup |U (x)| ] ≤ C1 .
(5.15)
x∈S
p − 2γ there exists a constant C > 0 which depends Then for each λ ∈ 0, p only on p, γ, λ, C0 and C1 such that sup εγ/λ N (ε, p; U ) ≤ C.
(5.16)
ε>0
Proof. Define F : Sp (U ) × Ω −→ R by F (X, ω) = X(ω). Then we have (5.17)
p p p E[|F (X) − F (Y )| ] = E[|X − Y | ] = dp (X, Y )
for any X, Y ∈ Sp (U ). By (5.14), (5.17) and the similar arguments in the proof of Theorem 1.4.1 in [7], we see that there exist a continuous modification F˜ of F and a constant C2 > 0 depending only on p, γ, λ and C0 such that F˜ (X) − F˜ (Y ) p ≤ C2 . sup E λ d (X, Y ) p X,Y ∈Sp (U ),00
ε>0
This implies our assertion. By Theorem 6, we can check [A4] under the following condition [A4 ]. ˜n (ε, M ; U ) [A4 ] For some γ2 ∈ (0, p0 /2), (1.6)−(1.10) hold with γ2 and N ˜ instead of γ0 and Nn (ε, M ; U ), where Nn (ε, M ; U ) is the smallest integer m m d such that there exist sets S1 , . . . , Sm which satisfy CM = Si and i=1
sup E n [|U (x) − U (y)|p0 ]1/p0 < ε
x,y∈Si
for each i = 1, . . . , m. 5.3. Examples In this section, we give two examples of Theorem 2. Let (Ω, F, P ) be 1 a complete probability space and let ξk = (ξki )m i=1 , k ∈ Z+ , be an m1 dimensional stationary Gaussian process. (a.)
2 Let f (x) = (f i (x))di=1 : Rm2 −→ Rd , u(t, x, y) = (ui (t, x, y))m i=1 :
Limits of Functional Stochastic Difference Equations
279
m1 3 [0, ∞) × Rd × Rm3 −→ Rm2 and ψ(x) = (ψ i (x))m −→ Rm3 be Borel i=1 : R 2 measurable functions. Let Ψ(t, w, y) = (Ψi (t, w, y))m i=1 and h(t, w, y) = (hi (t, w, y))di=1 be such that t i Ψ (t, w, y) = ui (s, w(t − s), ψ(y))ds 0
and hi (t, w, y) = f i (Ψ(t, w, y)). We define Fkn,i (w) and Gn,i k (w) by i Gn,i k (w) = E[h (k/n, w, ξk )]
(5.22) and
Fkn,i (w) = hi (k/n, w, ξk ) − Gn,i k (w).
(5.23)
We introduce the following conditions. [C1] f i (x) is three times continuously differentiable in x. Moreover u(t, x, y) is three times continuously differentiable in x and y, and all derivatives are continuous in t. [C2] It holds that
(5.24)
(5.25)
|β|+|β |≤2 0
|β|≤3 ∞
sup |Dβ f i (x)| < ∞,
x∈Rm2
sup
x∈Rd ,y∈Rm3
|Dxβ Dyβ uj (t, x, y)|dt < ∞
and sup |ψ ν (x)| < ∞
(5.26)
x∈Rm1
for each i = 1, . . . , d, j = 1, . . . , m2 and ν = 1, . . . , m3 . [C3] Let Gk,l = σ(ξνi ; i = 1, . . . , d, k ≤ ν ≤ l) and βk = sup sup{|P (A ∩ B) − P (A)P (B)| ; A ∈ G0,l , B ∈ Gk+l,∞ }. l
280
Takashi Kato
Then for some (4 ∈ (0, 1/2) ∞
(5.27)
k=1
4 1 < ∞. log(1/βk )
Define ˆbi (t, w) and ηkij (t, w) by ˆbi (t, w) = E[hi (t, w, ξ0 )]
(5.28) and (5.29)
ηkij (t, w) = E[hi (t, w, ξk )hj (t, w, ξ0 )] − ˆbi (t, w)ˆbj (t, w),
and a ˆij (t, w) by (5.30)
a ˆij (t, w) = η0ij (t, w) +
∞
ηkij (t, w) + ηkji (t, w) .
k=1
Let d d ∂2 ˆ f (t, w) = 1 ˆbi (t, w) ∂ f (w(t)) (5.31) a ˆij (t, w) i j f (w(t)) + 2 ∂x ∂x ∂xi i,j=1
i=1
for f ∈ C 2 (Rd ). Theorem 7. Assume [C1] − [C3]. Then the conclusion of Theorem 1 ˆ. holds replacing with Proof. We will check that Fkn,i and Gn,i k satisfy the assumptions of Theorem 2. [A1] − [A3], [B5] and [A6] are obvious. Proposition 16. The condition [B4] holds with γ1 = 1. Proof. Let U (w, ω) = hi (t, w, ξk (ω)). We define g(v, ω) : CˆR × Ω −→ R by g(v, ω) = f i (v(ψ(ξk (ω)))), where CˆR = C(KR ; Rm1 ), KR = {x ∈ m3 ˜ w, y) = sup |ψ i (x)|. We also define Ψ(t, Rm3 ; |x| ≤ R} and R = m1 i=1 x∈R
d ˜ j (t, w, y))m2 : [0, ∞) × CM (Ψ × KR −→ Rm2 by j=1 t ˜ j (t, w, y) = uj (s, w(t − s), y)ds. Ψ 0
Limits of Functional Stochastic Difference Equations
281
Then it follows that ˜ w, ·), ω). U (w, ω) = g(Ψ(t,
(5.32)
By [C2], we see that there is a constant C0 > 0 such that (5.33)
m2
d ˜ j (t, w, y)| ≤ C0 , w ∈ CM |Dyβ Ψ , y ∈ KR .
j=1 |β|≤1
Then we have d ˜ w, ·) ∈ AR , w ∈ CM , Ψ(t,
(5.34) where AR =
v ∈ CˆR ; v is continuously differentiable and m2
sup |Dβ v j (y)| ≤ C0 .
j=1 |β|≤1 |y|≤R
[C2] also implies (5.35)
|∇g(v, ω; v˜)| ≤ C1
m2
sup |˜ v j (y)|, v, v˜ ∈ AR , ω ∈ Ω
j=1 |y|≤R
for some C1 > 0. Then, by Theorem 5, we get (5.36)
ˆ (ε/C1 ; AR , d∞ ) N (ε, p, M ; U ) ≤ N
for each M > 0 and p ≥ 1, where d∞ (v, v ) = sup |v(y) − v (y)| and y∈KR
d . N (ε, p, M ; U ) is the minimum of cardinals of (ε, p, U )-nets of CM Moreover, by Theorem XIII in [8], we have
(5.37)
ˆ (ε/C1 ; AR , d∞ ) ≤ C1 C2 ε−1 log N
for some C2 > 0 depending only on R and C0 . Then we get (5.38)
log N (ε, p, M ; U ) ≤ C3 ε−1
for some C3 > 0 with U (w, ω) = hi (t, w, ξk (ω)).
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Takashi Kato
Similarly we see that (5.38) holds with U (w, ω) = ∇w hi (t, w, ξk (ω); Iln ej ) and U (w, ω) = ∇2w hi (t, w, ξk (ω); Iln ej , Iln eν ). Then we obtain the assertion. To check the condition [A7], we will show the following proposition. Proposition 17. For each K ∈ Kd , t ≥ 0 and k ∈ Z+ , it holds that [nt] + k [nt] i (5.39) ,y sup ,w · ∧ Ψ n n w∈K,y∈Rm1 −Ψi (t, w, y) −→ 0, n → ∞.
Proof. Let δT (s; w) = sup{|w(r) − w(r )| ; 0 ≤ r, r ≤ T, |r − r | ≤ s}, s, T > 0, w ∈ C([0, ∞); R). Then we have
[nt] + k [nt] i i sup , y − Ψ (t, w, y) ,w · ∧ Ψ m n n w∈K,y∈R 1 ([nt]+k)/n sup |ui (s, x, y)|ds ≤ t
d
x,y
∂ sup j ui (s, x, y) ∂x x,y j=1 0 [nt] + k
[nt] × sup wj −s ∧ − wj (t − s)ds n n w∈K ([nt]+k)/n sup |ui (s, x, y)|ds ≤ +
t
+
d j=1
0
t
x,y
t
∂ k + 1
sup j ui (s, x, y)ds sup δt ; wj . n x,y ∂x w∈K
Since K is compact, we see that k + 1
sup δt (5.40) ; wj −→ 0, n → ∞, k ∈ Z+ . n w∈K
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283
Then we have the assertion. n,i n,ij Define an,ij (k, w) and B n,ij (k, w) as in [A7]. 0 (k, w), b0 (k, w), A
Proposition 18. It holds that ij (i) sup |an,ij 0 ([nt], w) − η0 (t, w)| −→ 0, w∈K
ˆi (ii) sup |bn,i 0 ([nt], w) − b (t, w)| −→ 0, w∈K
(iii) sup |An,ij ([nt], w) − Aˆij (t, w)| −→ 0, w∈K
(iv) sup |B n,ij ([nt], w)| −→ 0 w∈K
for each t ≥ 0 and K ∈ K , where Aˆij (t, w) = d
∞
ηkij (t, w).
k=1
Proof. By Proposition 17, we get i i E[ sup |h ([nt]/n, w, ξk ) − h (t, w, ξk )|]
≤
w∈K m 2
∂ sup j f i (x) ∂x x j=1 [nt] [nt] j , y − Ψj (t, w, y) −→ 0 ×E sup ,w · ∧ Ψ m n n 1 w∈K,y∈R
as n → ∞. Then we have the assertion (ii). Moreover this implies ij sup |an,ij 0 ([nt], w) − η0 (t, w)| w∈K ≤ 2 sup |f i (x)| E[ sup |hj ([nt]/n, w, ξk ) − hj (t, w, ξk )|] x
w∈K
+ sup |f (x)| E[ sup |hi ([nt]/n, w, ξk ) − hi (t, w, ξk )|] −→ 0, n → ∞. j
x
w∈K
Then the assertion (i) holds. Since ξk is stationary, we have (5.41)
An,ij ([nt], w) =
∞ l=1
ηˆln,ij ([nt], w),
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Takashi Kato
where
k + l k k ηˆln,ij (k, w) = E hi , w · ∧ , ξl hj , w, ξ0 n n n
k
k + l k − E hi , w · ∧ , ξl E hj , w, ξ0 . n n n By Proposition 17, we have ηkn,ij ([nt], w) − ηkij (t, w)| sup |ˆ
w∈K m2
∂ i sup ν f (x) sup |f j (x)| ≤ 2 ∂x x x ν=1 [nt] + k [nt] ν ,w · ∧ , y − Ψν (t, w, y) × sup Ψ n n w∈K,y∈Rm2 + sup |f i (x)| E[ sup |hj ([nt]/n, w, ξ0 ) − hj (t, w, ξ0 )|] x
w∈K
−→ 0, n → ∞ for each k ∈ Z+ and t ≥ 0. Moreover, using Lemma 1, we have sup |ˆ ηkn,ij ([nt], w) − ηkij (t, w)| ≤ 16 sup |f i (x)| sup |f j (x)|βk ,
(5.42)
x
w∈K
x
and [C3] implies ∞
(5.43)
βk < ∞.
k=1
Thus the dominated convergence theorem implies sup |An,ij ([nt], w) − Aˆij (t, w)|
(5.44) ≤
w∈K ∞
sup |ˆ ηkn,ij ([nt], w) − ηkij (t, w)| −→ 0, n → ∞.
k=1 w∈K
This implies the assertion (iii). Since [nt] + k [nt]
n , y; I[nt] ∇ w hi ej ,w · ∧ n n m2 [nt] + k [nt]
∂ i ,y = Ψ f ,w · ∧ ∂xν n n ν=1
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285
[nt] ∂ ν [nt] + k [nt] + k n , y I[nt] × u ,w −s ∧ j ∂x n n n 0 [nt] + k
× − s ds, n k/n
we have (5.45)
∂ sup ν f i (x) sup |f j (x)| ∂x x x ν=1 ∞ k/n ∂ ν sup j u (s, x, y)dsβk . × ∂x x,y 0
sup |B n,ij ([nt], w)| ≤ 8
w∈K
m2
k=1
Then, [C2], (5.43) and the dominated convergence theorem imply the assertion (iv). By Proposition 18, we see that [A7] holds. Obviously a ˆij and ˆbi satisfies the condition [A8] and [A10]. Then, using Theorem 3, we obtain Theorem 7. 2 (b.) Let f (x) = (f i (x))di=1 : Rm2 −→ Rd , u(t, x, y) = (ui (t, x, y))m i=1 : m 3 [0, ∞) × Rm3 × Rm1 −→ Rm2 , and ψ(t, x) = (ψ i (t, x))i=1 : [0, ∞) × Rd −→ m3 2 R be Borel measurable functions. Let Ψ(t, w, y) = (Ψi (t, w, y))m i=1 and h(t, w, y) = (hi (t, w, y))di=1 be such that
i
Ψ (t, w, y) = 0
t
t
u s, ψ(r, w(r))dr, y ds i
s
and hi (t, w, y) = f i (Ψ(t, w, y)). We define Fkn,i (w) and Gn,i k (w) by (5.22) and (5.23). We introduce the following conditions. [D1] f i (x) is three times continuously differentiable in x. Moreover u(t, x, y) (respectively, ψ i (t, x)) is three times (respectively, twice) continuously differentiable in x, and all derivatives are continuous in t.
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Takashi Kato
[D2] It holds that
(5.46) (5.47)
|β|≤3 ∞
|β|≤2 0
and
x∈Rm2
sup
x∈Rm3 ,y∈Rm1
(5.48)
sup |Dβ f i (x)| < ∞,
|β|≤2 0
∞
|Dxβ uj (t, x, y)|dt < ∞
sup |Dxβ ψ ν (t, x)|dt < ∞
x∈Rd
for each i = 1, . . . , d, j = 1, . . . , m2 and ν = 1, . . . , m3 . Theorem 8. Assume [D1], [D2] and [C3]. Then the conclusion of ˆ which is defined by (5.28)-(5.31). Theorem 1 holds replacing with Theorem 8 is obtained by the similar arguments in the proof of Theorem 7. So we will check only the condition [B4]. Proposition 19. The condition [B4] holds with γ1 = 1. Proof. Let U (w, ω) = hi (t, w, ξk (ω)) and C˜t = C([0, t]; Rm3 ). We d 3 ˜ ˜ define ϕ(w) = (ϕj (w))m j=1 : C([0, ∞); R ) −→ Ct and g(v, ω) : Ct × Ω −→ R by t
j ψ j (r, w(r))dr ϕ (w) (s) = s
and g(v, ω) = f i
t
u s, v(s), ξk (ω) ds .
0
Then it follows that (5.49)
U (w, ω) = g(ϕ(w), ω).
Set C0 =
m3 j=1 |β|≤1 0
∞
sup |Dxβ ψ j (s, x)|ds.
x∈Rd
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287
By [D2], we see that C0 is finite and ϕ(w) ∈ A˜t , w ∈ C([0, ∞); Rd ),
(5.50) where A˜t =
v ∈ C˜t ; v is continuously differentiable and m3 j=1
d sup |v j (s)| + sup v j (s) ≤ C0 . 0≤s≤t 0≤s≤t ds
Moreover we have (5.51)
|∇g(v, ω; v˜)| ≤ C1
m3
sup |˜ v j (s)|, v, v˜ ∈ C˜t , ω ∈ Ω
j=1 0≤s≤t
for some C1 > 0. Then we have the assertion by the same arguments in the proof of Proposition 16. References [1] [2] [3] [4] [5] [6] [7] [8]
[9]
Billingsley, P., Convergence of probability measures 2nd. edition, WileyInterscience (1999). Borodin, A. N., A limit theorem for solutions of differential equations with random right-hand side, Theor. Prob. Appl. 22 (1977), 482–497. Fujiwara, T., Limit theorems for random difference equations driven by mixing processes, J. Math. Kyoto Univ. 32 (1992), 763–795. Ibragimov, I. A. and Yu. V. Linnik, Independent and stationary sequences of random variables, Groningen : Wolters-Noordhoff (1971). Kesten, H. and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys. 65 (1979), 97–128. Khasminskii, R. Z., A limit theorem for solutions of differential equations with a random right-hand sides, Theor. Prob. Appl. 11 (1966), 390–406. Kunita, H., Lecture on stochastic flows and applications, Tata Institute of Fundamental Research (1986). Kolmogorov, A. N. and V. M. Tihomirov, ε-entropy and ε-capacity of sets in functional spaces, Uspehi Mat. Nauk 14 (1959), 3–86, English transl., Amer. Math. Soc. Transl. (2) 17 (1961), 277–364. Kushner, H. J. and Hai-Huang, On the weak convergence of a sequence of general stochastic difference equations to a diffusion, SIAM J. Appl. Math. 40 (1981), 528–541.
288
[10]
[11]
[12] [13] [14] [15]
[16] [17]
Takashi Kato
Papanicolaou, G. C. and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math. 27 (1974), 641– 668. Papanicolaou, G. C., Stroock, D. W. and S. R. S. Varadhan, Martingale approach to some limit theorems, in Statistical Mechanics and Dynamical Systems, Duke Univ. Conf. Turbulence, Duke Univ. Math. Ser. 3, Durham, N.C. (1977). Schwartz, J. T., Nonlinear functional analysis, Gordon and Breach Science Publishers (1969). Stroock, D. W. and S. R. S. Varadhan, Diffusion processes with continuous coefficients I, Comm. Pure Appl. Math. 22 (1969), 345–400. Stroock, D. W. and S. R. S. Varadhan, Multidimensional diffusion processes, Springer-Verlag, Berlin (1979). Watanabe, H., Diffusion approximations of some stochastic difference equations, Pinsky, ed., Stochastic Analysis and Application, Marcel Dekker, New York (1984). Watanabe, H., Diffusion approximations of some stochastic difference equations II, Hiroshima Math. J. 14 (1984), 15–34. Watanabe, H., Diffusion approximations of some stochastic difference equations revisited, Stoch. Proc. Appl. 29 (1988), 147–154. (Received February 14, 2005) Graduate School of Mathematical Sciences The University of Tokyo Meguro-ku, Komaba 3-8-1 Tokyo 153-8914, Japan
