Malliavin calculus for stochastic diﬀerential equations with deterministic

We ﬁx T > 0. Let r be a positive integer, d₁, d₂, . . . , d_r positive integers, ϕ₁, ϕ₂, . . . , ϕ_r right-continuous increasing functions on [0, T] starting at 0. Set

W_k := {w; wis R^d^k-valued continuous function on [0, ϕ_k(T)], w(0) = 0},

Hk := {h∈C([0, ϕk(T)];R^d^k); h is absolutely continuous and ˙h∈L²([0, ϕk(T)];R^d^k)}, and let µ_k be Wiener measure on W_k for k= 1,2, . . . , r. We deﬁne the probability space (W, P) by

W := W₁×W₂×. . .×W_r, P := µ₁⊗µ₂⊗. . .⊗µ_r.

If we set

H :=H1⊗H2⊗. . .⊗Hr,

then (W, H, P) is an abstract Wiener space. Let (B_k(t)) be the canonical d_k-dimensional Brownian motion associated with (W_k, H_k, µ_k) for k= 1,2, . . . , r. Clearly, B₁, B₂, . . . , B_r are independent under P.

Next we consider stochastic diﬀerential equations with deterministic time change. Let Z_k(t) := B_k(ϕ_k(t)) for t ∈ [0, T] and k = 1,2, . . . , r and Ft the σ-ﬁeld generated by (Z_k(s); 0 ≤ s ≤ t, k = 1,2, . . . , r). Then, Z_k is a square-integrable (Ft)-martingale for every k = 1,2, . . . , r. We consider the following N-dimensional stochastic diﬀerential

equation: 





dX(t) =

∑r k=1

σ_k(t, X(t−))dZ_k(t) +b(t, X(t))dt, X(0) =x₀,

(4.2.1)

whereσkis anR^d^k⊗R^N-valued continuous function on [0, T]×R^N fork= 1,2, . . . , r,b is anR^N-valued continuous function on [0, T]×R^N, andx0 ∈R^N. We assume the estimate

maxk |σk(t, x)−σk(t, y)|+|b(t, x)−b(t, y)|< K|x−y|, x, y ∈R^N, t∈[0, T], max

k |σ_k(t, x)|+|b(t, x)|< K(1 +|x|), x∈R^N, t∈[0, T] with a positive constantK. Then we have the following theorem.

Theorem 4.2.1 The equation (4.2.1) has a unique (Ft)-adapted solution X = (X(t)) satisfying for any p >1

E [

sup

0≤t≤T

|X(t)|^p ]

≤x₀exp {

M (

T +

∑r k=1

ϕ_k(T) )}

, (4.2.2)

where M is a constant depending on r, p and K.

Proof. It is suﬃcient to show (4.2.2) in the case of p≥2. We use Picard’s successive approximation. Let (Ft)-adapted right-continuous processes {X_n} with left limits be deﬁned by

X₀(t) := x₀, X_n+1(t) := x₀+

∫ t 0

∑r k=1

σ_k(s, X_n(s−))dZ_k(s) +

∫ t 0

b(s, X_n(s))ds.

It is to be noted that the discontinuous points of X_n correspond with the discontinuous points ofϕfor anynalmost surely. Now we show that there exists a constantM depending

onp and K such that E

[ sup

0≤s≤t|X_n+1(s)−X_n(s)|^p ]¹_p

≤ x₀ 2ⁿ exp

{ M

( t+

∑r k=1

ϕ_k(t) )}

(4.2.3) by induction on n. When n = 1, it is easy to see that the inequality (4.2.3) holds for suﬃciently large M. We assume the inequality (4.2.3) forn−1. Lemma 4.1.3 leads to

E [

sup

0≤s≤t|X_n+1(s)−X_n(s)|^p ]¹

≤

∑r k=1

E [

sup

0≤s≤t

¯¯¯¯∫ s 0

(σ_k(u, X_n(u−))−σ_k(u, X_n₋₁(u−)))dZ_k(u)¯¯

¯¯^p]¹_p +E

[ sup

0≤s≤t

¯¯¯¯∫ s 0

(b(u, X_n(u))−b(u, X_n₋₁(u)))du¯¯

¯¯^p]¹_p

≤C₃(p)





∑r k=1

E [(∫ t

|σ_k(u, X_n(u−))−σ_k(u, X_n₋₁(u−))|²dϕ_k(u) )^p₂]¹_p

+E[¯¯

¯¯∫ _t

(b(u, X_n(u))−b(u, X_n₋₁(u)))du¯¯

¯¯^p]¹_p}

≤C₄(p, K)





∑r k=1

[(∫ t 0

|X_n(u−)−X_n₋₁(u−)|²dϕ_k(u) )^p₂]¹_p

+E [(∫ _t

|X_n(u)−X_n₋₁(u)|du

)p]¹_p}

≤C4(p, K) { _r

∑

k=1

(∫ t 0

E[|Xn(u−)−Xn−1(u−)|^p]²^pdϕk(u) )¹₂

∫ t 0

E[|Xn(u)−Xn−1(u)|^p]¹^pdu }

. From the assumption of induction we derive

E [

sup

0≤s≤t|X_n+1(s)−X_n(s)|^p ]¹

≤ x₀

2ⁿC₄(p, K)





∑r k=1

(∫ t 0

exp {

2M (

∑r k=1

ϕ_k(u−) )}

dϕ_k(u) )¹₂

∫ _t

exp {

M (

∑r l=1

ϕ_l(u) )}

du }

≤ x0

2ⁿC4(p, K) [ _r

∑

k=1

exp {

M (

t+∑

l̸=k

ϕl(t−)

)} (∫ t 0

exp (2M ϕk(u−))dϕk(u) )¹₂

+ exp (

∑r l=1

ϕ_l(t) ) ∫ t

e^{M u}du ]

. Since ϕ_k(ϕ⁻_k¹(s)−)≤s, Lemma 4.1.1 implies

∫ t 0

exp (2M ϕ_k(u−))dϕ_k(u) =

∫ ϕk(t) 0

exp(

2M ϕ_k(ϕ⁻¹_k (s)−)) ds

≤

∫ ϕk(t) 0

exp(2M s)ds

≤ 1

2M exp(2M ϕ_k(t)).

Hence it is holds that E

[ sup

0≤s≤t|X_n+1(s)−X_n(s)|^p ]¹_p

≤ x₀

2ⁿC₄(p, K) [ _r

∑

k=1

exp {

M (

t+∑

l̸=k

ϕ_l(t−) )}

√1

2Me^{M ϕ}^k^(t) + exp

( M

∑r l=1

ϕ_l(t) )

1 Me^{M t}

]

≤ x₀

2ⁿC₄(p, K) { r

√2M + 1 M

} exp

{ M

( t+

∑r l=1

ϕ_l(t) )}

. If we choose M suﬃciently large such that

( r

√2M + 1 M

)

C₄(p, K)≤ 1 2,

then the inequality (4.2.3) holds for n+ 1. Therefore we complete the induction.

The inequality (4.2.3) leads to E

[ sup

0≤s≤t|X_n(s)−X_m(s)|^p ]_p¹

≤ x₀ 2^m exp

{ M

( t+

∑r k=1

ϕ_k(t) )}

for any positive integers n and m such that n > m. This inequality implies that {X_n} is a Cauchy sequence. Hence there exists an (Ft)-adapted right-continuous process X with left limits such that the discontinuous points of X correspond with the discontinuous points of ϕ almost surely, and

nlim→∞E [

sup

0≤s≤t|X(s)−Xn(s)|^p ]¹_p

= 0.

It is easily seen that X is a solution of the equation (4.2.1). The estimate of X follows easily.

To prove the uniqueness, let X and Y be two solutions of the equation (4.2.1). A similar discussion as above yields

E [

sup

0≤s≤t|X(s)−Y(s)|² ]

≤C₅(p, r, K) { _r

∑

k=1

∫ _t

E[|X(s−))−Y(s−)|²]dϕ_k(s) +

∫ _t

E[|X(s))−Y(s)|²]ds }

≤C₅(p, r, K)

∫ _t

E [

sup

0≤u≤s|X(u−))−Y(u−)|² ]

d (

∑r k=1

ϕ_k(s) )

Applying Lemma 4.1.4 to this inequality, we have E

[ sup

0≤t≤T|X(t)−Y(t)|² ]

= 0.

Now we apply Malliavin calculus to the solution X = (X(t)) of the equation (4.2.1).

Theorem 4.2.2 We assume thatσk∈C^0,m([0, T]×R^N;R^d^k⊗R^N)and∇σk∈C_b^0,m⁻¹([0, T]× R^N;R^N⊗R^d^k⊗R^N)fork = 1,2, . . . , r, b∈C^0,m([0, T]×R^N;R^N), and∇b ∈C_b^0,m⁻¹([0, T]× R^N;R^N ⊗R^N). Then we have X(t) ∈ W^m,p(R^N) for t ∈ [0, T], and there exists a con-stant M depending on r, p, m and the bounds of the spatial derivatives of σk and b up to order m such that

||X(t)||m,p ≤exp {

M (

∑r k=1

ϕ_k(t) )}

, t ∈[0, T]. (4.2.4) Proof. It is suﬃcient to show (4.2.4) in the case p ≥2. We deﬁne X_n as in the proof of Theorem 4.2.1. The proof of Theorem 4.2.1 implies that X_n(t) converges to X(t) in L^p for eacht ∈[0, T]. Now we proceed to show thatX_n(t) is inW^m,p for everyt ∈[0, T] and alln, and satisﬁes that the inequality

E [

sup

0≤s≤t|D^jXn(s)|^p_L^j

2(H;R^N)

]¹_p

≤exp {

M (

∑r k=1

ϕk(t) )}

(4.2.5) with a constantM depending on p, mand the bounds of the spatial derivatives of σ_k and b up to order msuch that for n= 0,1,2, . . . , j = 0,1, . . . , m. For it we use the induction on (n, j). By the proof of Theorem 4.2.1, we know thatX_n(t) is in L^p for eacht ∈[0, T]

and all n, and there exists a constant M such that (4.2.5) holds for j = 0. Clearly X0(t) is in W^m,p for each t ∈ [0, T], and there exists a constant M such that (4.2.5) holds for n = 0. Let j0 ≤ m. Next, as the hypothesis of the induction we assume that Xn(t) is in W^j,p for “each t ∈ [0, T], all n and j = 0,1, . . . , j0 −1” and for “each t ∈ [0, T], n= 0,1, . . . , n0 and j = 0,1, . . . , j0”, and that there exists a constantM satisfying (4.2.5) for “all n and j = 0,1, . . . , j0 −1” and for “n = 0,1, . . . , n0 and j = 0,1, . . . , j0”. Then we show that Xn0+1(t) is in W^j⁰^,p for each t ∈ [0, T], and that there exists a constant M satisfying (4.2.5) for n0 + 1 and j0. Proposition 4.1.5 gives the explicit expression of DXn0+1

DX_n₀₊₁(t) =

∑r k=1

∫ _t

∇σ_k(s, X_n₀(s−))DX_n₀(s−)dZ_k(s) +

∫ _t

∇b(s, X_n₀(s))DX_n₀(s)ds

+∑

∑r k=1

h^λ_k⊗

∫ _ϕ_k_(t)

σ_k(ϕ⁻_k¹(s), X_n₀(ϕ⁻_k¹(s))) ˙h^λ_k(s)ds,

where {h_λ = (h^λ₁, h^λ₂, . . . , h^λ_r)}λ is a complete orthonormal normal system of H = H₁⊗ H₂⊗. . .⊗H_r. Repeating this procedure, we have

D^j⁰X_n₀₊₁(t)

∑r k=1

∫ _t

{∇σ_k(s, X_n₀(s−))D^j⁰X_n₀(s−)

∑

l=1

A^k_l(s, X_n₀(s−))Q^k_l(DX_n₀(s−), . . . , D^j⁰⁻¹X_n₀(s−)) }

dZ_k(s) +

∫ t 0

{∇b_k(s, X_n₀(s))D^j⁰X_n₀(s)

∑

l=1

A˜^k_l(s, X_n₀(s)) ˜Q^k_l(DX_n₀(s), . . . , D^j⁰⁻¹X_n₀(s)) }

+∑

∑r k=1

h^λ_k ⊗

∫ ϕk(t) 0

j∑0−1 l=0

Aˆ^k_l(ϕ⁻_k¹(s), Xn0(ϕ⁻_k¹(s)))

×Qˆ^k_l(DXn0(ϕ⁻_k¹(s)), . . . , D^j⁰⁻¹Xn0(ϕ⁻_k¹(s))) ˙h^λ_k(s)ds,

(4.2.6) where A^k_l,A˜^k_l ∈ C_b¹([0, T]×R^N; (R^N)^⊗^l ⊗R^d^k ⊗R^N) , ˆA^k_l ∈ C¹([0, T]×R^N; (R^N)^⊗^l⊗ R^d^k⊗R^N) satisfy that

maxl,k |Aˆ^k_l(t, x)| ≤C6({||∇^lσk||∞}1≤l≤m,1≤k≤r)(1 +|x|), x∈R^N, t∈[0, T],

and Q^k_l,Q˜^k_l,Qˆ^k_l are (R^N)^⊗^l⊗H^j⁰-valued functions whose components are polynomials of

order l. Therefore, from Lemma 4.1.3, it follows that E

[ sup

0≤s≤t|D^j⁰X_n₀₊₁(s)|^p_L^j0 2 (H;R^N)

]¹

≤C₇(p)E [( _r

∑

k=1

∫ t 0

|∇σ_k(s, X_n₀(s))D^j⁰X_n₀(s)

∑

l=1

A^k_l(s, X_n₀(s))Q^k_l(DX_n₀(s), . . . , D^j⁰⁻¹X_n₀(s))|²_L^j0

2 (H;R^N)dϕ_k(s) )^p₂



1 p

∫ _t

|∇b_k(s, X_n₀(s))D^j⁰X_n₀(s)

∑

l=1

A˜^k_l(s, X_n₀(s)) ˜Q^k_l(DX_n₀(s), . . . , D^j⁰⁻¹X_n₀(s))|^p

L2^j⁰(H;R^N)

]¹_p ds

∑r k=1

[(∫ ϕk(t) 0

j∑0−1 l=0

Aˆ^k_l(ϕ⁻_k¹(s), X_n₀(ϕ⁻_k¹(s)))

×Qˆ^k_l(DX_n₀(ϕ⁻_k¹(s)), . . . , D^j⁰⁻¹X_n₀(ϕ⁻_k¹(s)))|²_L^j0

2 (H;R^N)ds )^p₂



1 p

. From Lemma 4.1.1 the last term of this inequality equals to

∑r k=1



(∫ t 0

j∑0−1 l=0

Aˆ^k_l(s, X_n₀(s−)) ˆQ^k_l(DX_n₀(s−), . . . , D^j⁰⁻¹X_n₀(s−))|²_L^j0

2 (H;R^N)dϕ_k(s) )^p₂



1 p

On the other hand, the induction assumptions imply that E

[ sup

0≤s≤t|D^jX_n₀(s)|^p_L^j

2(H;R^N)

]¹_p

≤C₈(

m, p,{||∇^lσ_k||∞}1≤l≤m,1≤k≤r,{||∇^lb||∞}1≤l≤m

)

×exp {

C₈(

m, p,{||∇^lσ_k||_∞}1≤l≤m,1≤k≤r,{||∇^lb||_∞}1≤l≤m

)( t+

∑r k=1

ϕ_k(t) )}

, j = 0,1, . . . , j₀. Hence, by H¨older’s inequality, we have

[|A^k_l(s, X_n₀(s))Q^k_l(DX_n₀(s), . . . , D^j⁰⁻¹X_n₀(s))|^p_L^j0 2 (H;R^N)

]

≤C₉(

m, p,{||∇^lσ_k||∞}1≤l≤m,1≤k≤r,{||∇^lb||∞}1≤l≤m

)

×exp {

C₉(

m, p,{||∇^lσ_k||_∞}1≤l≤m,1≤k≤r,{||∇^lb||_∞}1≤l≤m

)( t+

∑r k=1

ϕ_k(t) )}

The same estimates also hold for ˜A^k_lQ˜^k_l and ˆA^k_lQˆ^k_l. Then we can make similar argument to the proof of Theorem 4.2.1, so that Xn0+1(t) ∈W^j⁰^,p and (4.2.5) holds for suﬃciently large M depending onr, p, m and the bounds of the spatial derivatives of σk and b up to order m. Thus we have

X_n(t)−→X(t) in L^p, sup

n ||X_n(t)||m,p<∞.

In help of Lemma 1.5.3. in [19], we have the conclusion. ¤ Next we consider the relation between the ellipticity of equations and the non-degeneracy of Malliavin covariance matrices.

Theorem 4.2.3 We assume that σ_k∈C^0,1([0, T]×R^N;R^d^k⊗R^N) and ∇σ_k is bounded for k = 1,2, . . . , r, b ∈ C^0,1([0, T]×R^N;R^N), ∇b is bounded, and that there exists a positive constant ε such that

∑r k=1

σ_k(0, x₀) ^tσ_k(0, x₀)≥ε.

Then, Malliavin covariance matrix∆(t) = ((DXⁱ(t), DX^j(t))_H∗)_ij is invertible, and there exists a constant C =C(x₀, N, p, ε, r,{||∇σ_k||∞}1≤k≤r,||∇b||∞) such that for all p >1 E[det(∆(t))⁻^p]≤Cmin{ϕ_i(t);i= 1,2, . . . , r}⁻^{N p}exp [C(t+ max{ϕ_i(t);i= 1,2, . . . , r})].

(4.2.7) Moreover, if there exists a positive constant ε and t₀ such that

∑r k=1

σ_k(t, x) ^tσ_k(t, x)≥ε, t ∈[0, t₀], x∈R^N,

then we can choose a constant C = C(t0, N, p, ε, r,{||∇σk||∞}1≤k≤r,||∇b||∞) satisfying (4.2.7).

Proof. Let

A_k(t) := [B_k(ϕ_k(·)), B_k(ϕ_k(·))](t), t∈[0, T], k = 1,2, . . . , r.

We deﬁne twoN ×N-matrix-valued processesJ₁ and J₂ as the solutions of the following stochastic diﬀerential equations, respectively.







dJ₁(t) =

∑r k=1

∇σ_k(t, X(t−))J₁(t−)dZ_k(t) +∇b(t, X(t−))J₁(t−)dt, J₁(0) =I,











dJ₂(t) =−

∑r k=1

J₂(t−)∇σ_k(t, X(t−))dZ_k(t)−J₂(t−)∇b(t, X(t−))dt +

∑r k=1

J₂(t−)∇σ_k(t, X(t−))∇σ_k(t, X(t−))dA_k(t), J₂(0) =I.

Corollary 2 and Theorem 29 of Section 6 of Chapter II in [20] imply that J₁(t)J₂(t) =I for all t ∈[0, T], from which it follows thatJ1(t) = J2(t)⁻¹ and

J₂(t)DX(t)[h] =

∑r k=1

∫ _t

J₂(s−)σ_k(s, X(s−))dh_k(ϕ_k(s)),

with h= (h₁, h₂, . . . , h_r)∈H. By virtue of Lemma 4.1.1 this can be expressed as J2(t)DX(t)[h] =

∑r k=1

∫ ϕk(t) 0

J2(ϕ⁻_k¹(u)−)σk(ϕ⁻_k¹(u), X(ϕ⁻_k¹(u)−)) ˙hk(u)du.

Hence, for a complete orthonormal normal system {h^λ} of H we have

∆(t) = J₁(t)∑

∑r k=1

∫ ϕ_k(t) 0

J₂(ϕ⁻_k¹(u)−)σ_k(ϕ⁻_k¹(u), X(ϕ⁻_k¹(u)−)) ˙h^λ_k(u)du

∫ ϕk(t) 0

t[J2(ϕ⁻_k¹(u)−)σk(ϕ⁻_k¹(u), X(ϕ⁻_k¹(u)−))] ˙h^λ_k(u)du^tJ1(t)

= J₁(t)

∑r k=1

∫ _ϕ_k_(t)

J₂(ϕ⁻_k¹(u)−)σ_k(ϕ⁻_k¹(u), X(ϕ⁻_k¹(u)−))

×^tσ_k(ϕ⁻_k¹(u), X(ϕ⁻_k¹(u)−))^tJ₂(ϕ⁻_k¹(u)−)du^tJ₁(t)

= J₁(t)

∑r k=1

∫ _t

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s)^tJ₁(t).

From this one can derive det(∆(t)) = det(J₁(t))²det

( _r

∑

k=1

∫ _t

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s) )

. (4.2.8) For the estimate of det(J₁(t)), the following lemma holds.

Lemma 4.2.4 E[|det(J₂(t))|^p]

≤C₁₀(p, N, r,{||∇σ_k||∞}1≤k≤r,||∇b||∞)

×exp [C₁₀(p, N, r,{||∇σ_k||∞}1≤k≤r,||∇b||∞)(t+ max{ϕ_i(t);i= 1,2, . . . , r})].

Proof of Lemma 4.2.4. Lemma 4.1.3 enable us to make similar discussion in the proof of Theorem 4.2.1, and it follows

maxi,j E [

sup

0≤s≤t|(J₂(s))_ij|^p ]¹

≤C₁₁(p, r,{||∇σ_k||_∞}1≤k≤r,||∇b||_∞)

∫ t 0

maxi,j E [

sup

0≤u≤s|(J2(u))ij|^p ]¹_p

d (

∑r k=1

ϕk(s) )

. Lemma 4.1.4 yields that

maxi,j E[|(J₂(t))_ij|^p]¹^p ≤ C₁₁(p, r,{||∇σ_k||∞}1≤k≤r,||∇b||∞)

×exp [C11(p, r,{||∇σk||∞}1≤k≤r,||∇b||∞)

×(t+ max{ϕ_i(t);i= 1,2, . . . , r})]. By H¨older’s inequality, we have

E[|det(J₂(t))|^p]≤N! max

i,j E[|(J₂(t))_ij|^{N p}]^{N p}¹ . Therefore we have the conclusion of Lemma 4.2.4.

The estimate of Lemma 4.2.4 is suﬃcient for the part det(J1(t)). We estimate the other part. Let ξ ∈ S^N⁻¹ where S^N⁻¹ is the (N −1)-dimensional sphere centered at 0.

From the assumption of ellipticity and the compactness of S^N⁻¹, we can choose n ∈ N, open sets Gi inS^N⁻¹, and ki = 1,2, . . . , r, for i= 1,2, . . . , nsuch that

∪n i=1

G_i =S^N⁻¹,

tξσ_k_i(0, x₀)^tσ_k_i(0, x₀)ξ > ε

2r, ξ ∈G_i, i= 1,2, . . . , n.

By the continuity of {σ_k}, there exist R_i >0 andt_i ∈(0, T] such that

tξσ_k_i(s, x)^tσ_k_i(s, x)ξ > ε

3r, x∈B(x₀, R_i), s∈[0, t_i] , ξ∈G_i

for i= 1,2, . . . , n. Let R:= miniRi and t0 := miniti. We deﬁne a stopping time ζ by ζ := inf{t∈[0, T];|X(t)−x₀|> R or |J₁(t)−I|> δ} ∧T,

where δ∈(0, t₀) is chosen so small that

tξJ₂(s)σ_k_i(s, x)^tσ_k_i(s, x)^tJ₂(s)ξ≥ ε 4r,

x∈B(x₀, R), s∈[0, ζ), ξ ∈S^N⁻¹.

To simplify the notation, we denote min{ϕi(t);i= 1,2, . . . , r}by η(t). We note that η is also a right-continuous increasing function on [0, T]. Since Lemma 4.1.1 yields that for i= 1,2, . . . , n and ξ∈Gi

tξ ( _r

∑

k=1

∫ t 0

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s) )

≥

∫ t∧ζ 0

tξJ₂(s−)σ_k_i(s, X(s−))^tσ_k_i(s, X(s−))^tJ₂(s−)ξdϕ_k_i(s)

≥ ε

4rη(t∧ζ), we have

det ( _r

∑

k=1

∫ t 0

J2(s−)σk(s, X(s−))^tσk(s, X(s−))^tJ2(s−)dϕk(s) )

≥4⁻^Nr⁻^Nε^Nη(t∧ζ)^N. Hence

E [

det ( _r

∑

k=1

∫ _t

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s) )₋p]

≤4^{N p}r^{N p}ε^{−N p}E[η(t∧ζ)^{−N p}]

= 4^{N p}r^{N p}ε⁻^{N p}E[η(t)⁻^{N p};ζ ≥t] + 4^{N p}r^{N p}ε⁻^{N p}E[η(ζ)⁻^{N p};ζ < t].

Since η(η⁻¹(u)−)≤u, from Lemma 4.1.1 we have η(ζ)^{−N p}−η(t)^{−N p} = N p

∫ η(t) η(ζ)

u^{−N p−1}du

≤ N p

∫ η(t) η(ζ)

η(η⁻¹(u)−)⁻^{N p}⁻¹du

= N p

∫ t ζ

η(s−)^{−N p−1}dη(s).

Hence we have E

[ det

( _r

∑

k=1

∫ _t

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s) )₋p]

≤4^{N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p}P(ζ ≥t) +4^{N p}r^{N p}ε⁻^{N p}E

[ N p

∫ _t

η(s−)⁻^{N p}⁻¹dη(s) +η(t)⁻^{N p};ζ < t ]

= 4^{N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p}

+4^{N p}r^{N p}ε⁻^{N p}N pE [∫ t

1_(ζ,t](s)η(s−)⁻^{N p}⁻¹dη(s);ζ < t ]

= 4^{N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p} +4^{N p}r^{N p}ε^{−N p}N p

∫ t 0

η(s−)^{−N p−1}E[1_(ζ,t](s);ζ < t]dη(s)

= 4^{N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p}+ 4^{N p}r^{N p}ε⁻^{N p}N p

∫ t 0

η(s−)⁻^{N p}⁻¹P(ζ < s)dη(s). (4.2.9) On the other hand, we have the following estimate for ζ.

Lemma 4.2.5

P(ζ ≤t)≤2N rexp{−C₁₂(N, r, R, δ,||∇b||∞)η(t)⁻¹}.

Proof of Lemma 4.2.5. Note that it is suﬃcient to prove the estimate for smallt. Set ζ₁ = inf{t ∈[0, T];|X(t)−x₀|> R} ∧T,

ζ2 = inf{t ∈[0, T];|J1(t)−I|> δ} ∧T.

Then it is suﬃcient to prove the same estimate for ζ₁ and ζ₂. Since the estimates for ζ₁ and ζ₂ are proved similarly, we prove the estimate only for ζ₂. We deﬁne continuous martingale (M_k(t)) by

M_k(t) :=

∫ _t

∇σ_k(ϕ⁻_k¹(s), X(ϕ⁻_k¹(s)−))J₁(ϕ⁻_k¹(s)−)dB_k(s), k = 1,2, . . . , r.

Denoting ∑_N

i,j=1⟨(M_k)_ij⟩by ⟨M_k⟩ for k =,1,2, . . . , r, we have

⟨Mk⟩(ϕk(t∧ζ2)) =

∫ ϕk(t∧ζ2) 0

|∇σk(ϕ⁻_k¹(s), X(ϕ⁻_k¹(s)−))J1(ϕ⁻_k¹(s)−)|²ds

≤ C₁₃(δ)ϕ_k(t∧ζ₂).

From Lemma 4.1.2, it follows that sup

s∈[0,t]

|J₁(s∧ζ₂)−I| ≤

∑r k=1

sup

s∈[0,ϕk(t∧ζ2)]

|M_k(s)|+C₁₄(||∇b||∞)t.

Therefore, ift ≤ _2C₁₄₍_||∇^δ _b_||_∞₎, then by Proposition 6.8 of [24], we have P(ζ₂ ≤t) ≤ P

( sup

s∈[0,t]

|J₁(s∧ζ₂)−I| ≥δ )

≤ P ( _r

∑

k=1

sup

s∈[0,ϕ_k(t∧ζ2)]

|M_k(s)| ≥ δ 2

)

≤

∑r k=1

P (

sup

s∈[0,ϕk(t∧ζ2)]

|M_k(s)| ≥ δ 2r

)

∑r k=1

P (

sup

s∈[0,ϕk(t∧ζ2)]

|M_k(s)| ≥ δ

2r, ⟨M_k⟩(ϕ_k(t∧ζ₂))≤C₁₃(δ)ϕ_k(t∧ζ₂) )

= 2N

∑r k=1

exp (

− δ²

8N²r²C₁₃(δ)ϕ_k(t) )

This completes the proof of Lemma 4.2.5.

From Lemma 4.2.5 it holds that

P(ζ < t)≤2N rexp{−C₁₂(N, r, R, δ,||∇b||∞)η(t−)⁻¹}. Therefore, by (4.2.9) we have

E [

det ( _r

∑

k=1

∫ _t

J₂(s−)σ_k(s, X(s−))^tσ_k(s, X(s−))^tJ₂(s−)dϕ_k(s) )₋p]

≤2^{2N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p} +2^{2N p+1}r^{N p+1}ε⁻^{N p}N²p

∫ _t

η(s−)⁻^{N p}⁻¹exp{−C₁₂(N, r, R, δ,||∇b||∞)η(s−)⁻¹}dη(s).

Now we estimate the second term in the right hand side of above inequality. Let f(x) := x⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)x⁻¹.

It is easily seen that f is a positive, bounded, and concave function on (0,∞), and the maximum is attained at x = ^C¹²^(N,r,R,δ,_{N p+1}^||∇^b^||^∞⁾. We denote the maximum jump of (η(t);t ∈ [0, T]) by J. Since u−J ≤η(η⁻¹(u)−) ≤u, the following inequalities hold by Lemma 4.1.1

∫ t 0

η(s−)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)η(s⁻⁾⁻¹dη(s)

∫ _η(t)

η(η⁻¹(u)−)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)η(η⁻¹^(u)⁻⁾⁻¹du

≤

∫ _J+^C¹²⁽^N,r,R,δ,^||∇^b^||∞⁾

N p+1

η(η⁻¹(u)−)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)η(η⁻¹^(u)⁻⁾⁻¹du +

∫ _∞

J+^C¹²⁽^N,r,R,δ,_{N p+1}^||∇^b^||∞⁾

η(η⁻¹(u)−)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)η(η⁻¹^(u)⁻⁾⁻¹du

≤ ||f||_∞ (

J+ C₁₂(N, r, R, δ,||∇b||_∞) N p+ 1

)

∫ _∞

J+_{N p+1}¹

(u−J)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)(u⁻^J)⁻¹du

≤ ||f||∞

(

J+ C₁₂(N, r, R, δ,||∇b||_∞) N p+ 1

)

∫ _∞

u⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)u⁻¹du.

Changing variables in the integral of the right hand side, we have

∫ _∞

u⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)u⁻¹du=C₁₂(N, r, R, δ,||∇b||_∞)⁻^{N p}Γ(N p).

Here Γ is the gamma function. This leads to the inequality

∫ t 0

η(s−)⁻^{N p}⁻¹e⁻^C¹²^(N,r,R,δ,^||∇^b^||^∞^)η(s⁻⁾⁻¹dη(s)≤C₁₅(N, p, r, R, δ,||∇b||_∞)(1 +J).

Therefore, we can conclude that for all t∈[0, T] E

[ det

( _r

∑

k=1

∫ t 0

J2(s−)σk(s, X(s−))^tσk(s, X(s−))^tJ2(s−)dϕk(s) )₋p]

≤2^{2N p}r^{N p}ε⁻^{N p}η(t)⁻^{N p}+ 2^{2N p+1}r^{N p+1}ε⁻^{N p}N²pC₁₅(N, p, r, R, δ,||∇b||_∞)(1 +J)

≤C₁₆(N, p, r, ε, R, δ,||∇b||∞)(1 +J+η(t)⁻^{N p}).

Note that R and δ are determined by {||∇σ_k||_∞}1≤k≤r, x₀, and ε. From (4.2.8), this estimate, and Lemma 4.2.4, the ﬁrst assertion follows. Since the condition of the second assertion implies that the constants for the estimates can be chosen independently of x₀ but dependently on t₀, the second assertion is derived. ¤ Theorems 4.2.2 and 4.2.3 enable us to apply Sobolev’s inequality with respect to H-derivative to the solution of the stochastic diﬀerential equation (4.2.1), and the following theorem follows.

Theorem 4.2.6 For the stochastic diﬀerential equation (4.2.1), we assume that σ_k ∈ C^0,m+2([0, T]×R^N;R^d^k ⊗R^N), ∇σ_k ∈ C_b^0,m+1([0, T]×R^N;R^N ⊗R^d^k ⊗R^N) for k = 1,2, . . . , r, b∈C^0,m+2([0, T]×R^N;R^N), ∇b∈C_b^0,m+1([0, T]×R^N;R^N ⊗R^N), and there exists a positive constant ε such that

∑r k=1

σ_k(0, x₀) ^tσ_k(0, x₀)≥ε.

Then, the law P(t, x0, dy)of X(t, x0)is absolutely continuous with respect to the Lebesgue measure and its density function p(t, x, y) is estimated as

0max≤l≤m sup

y∈R^d

¯¯∇^lyp(t, x₀, y)¯¯≤c₁min{ϕ_i(t);i= 1,2, . . . , r}⁻^c³exp {

c₂ (

∑r k=1

ϕ_k(t) )}

. (4.2.10) with positive constants c₁, c₂, c₃. Moreover, if there exist positive constants ε and t₀ such

that ∑r

k=1

σk(t, x) ^tσk(t, x)≥ε, t ∈[0, t0], x∈R^N,

then the constants c₁, c₂, c₃ in (4.2.10) can be chosen independently of x₀ but dependently on t₀.

Proof. The conclusion follows from Theorems 4.2.3, 4.2.2, and Theorem 5.9 of [24]. ¤

ドキュメント内 Existence of Densities and their Regularity for Solutions of Stochastic Diﬀerential (ページ 46-60)