High order weak approximation for irregular functionals of time-inhomogeneous SDEs

A Proof of Lemma 3.7

For k∈ℕ and for multi-indices αe∈{0,1,…,d}re, re∈ℕ, e=1,…,k such that ∥αe∥≥2, e=1,…,k, we have

By Proposition 3.5, it holds that

By integration by parts, we have

Then we obtain the assertion.

B Proof of Lemma 3.8

By integration by parts, we have

The computation of HI⁢(X¯st,x,𝔹γ⋆,t,sSkor) is reduced to the computation of Skorohod integrals due to the following:

Then we obtain the assertion.

C Proof of Lemma 3.9

Instead of the standard integration by parts on the Wiener space, we apply a Bismut type formula when N=d:

for G=(G1,…,Gd)∈(𝔻∞)d. Iteratively using this formula, we obtain the assertion.

D Proof of Lemma 4.1

In the proof, we use the method of integration by parts with uniform non-degeneracy of the Malliavin covariance matrix in the sense of Watanabe. We introduce the scaling SDE with a parameter ε∈(0,1]:

with Xtt,x,(ε)=x∈ℝN. We define

which is expanded by using the Stratonovich–Taylor expansion (Kloeden and Platen (1999)) as

where re+1ε⁢(t,s,x) is the residual. Here, a family of Wiener functionals Fε⁢(t,s,x), ε∈(0,1], is uniformly non-degenerate and it holds that

We put F0⁢(t,s,x):=∑i=1dVi⁢(t,x)⁢Bt,si. By Taylor expansion and integration by parts, we have

where

with Wiener functionals Ebi,Iiη⁢(t,s,x)∈𝔻∞, i=1,…,k, satisfying that for ℓ∈ℕ, p>1,

Then it holds that

where

The estimate of the weight Rmη⁢(t,s,x) in the error term is reduced to the following: for I∈{1,…,N}k, b1,…,bk≥2 such that ∑i=1kbi=2⁢m+2+k, for p>1, there is C>0 such that for x∈ℝN, t<s and η∈[0,1],

by [9, Corollary 3.7]. Note that D⁢Fη⁢(t,s,x)=D⁢Xsη,t,x/η. The factor ∥D⁢Fη⁢(t,s,x)∥ℓ,q,H is bounded, for ℓ∈ℕ and q>1, by

by [9, Theorem 2.19]. Also, by [9, Theorem 3.5], for q>1 we have

Since for ℓ∈ℕ and q>1,

we have that for p>1 there exists C>0 such that

for t<s. Therefore, there is C>0 such that

for all φ∈Cb∞⁢(ℝN) and t<s.

E Proof of Lemma 4.2

By applying Lemma 3.7 and Lemma 3.8 with (2.3) to the expansion in Lemma 4.1, we have

where

and ℰt,s1⁢φ satisfies that there exists C>0 such that

If we use the smoothness of φ, the expectations of type E⁢[φ⁢(X¯st,x)⁢𝔹κ∗γ⋆,t,sSkor] with length |γ|≥m+1 in 𝒯t,s⁢φ⁢(x) will be of O⁢((s-t)m+1) due to the following identity: for |γ|≥m+1,

where the calculation in the proof of Lemma 3.7 and the duality formula (2.1) are applied. Then we define

i.e.

Here, in E⁢[φ⁢(X¯st,x)⁢𝒲(m)⁢(t,s,x,Bt,s)], expectations of type E⁢[φ⁢(X¯st,x)⁢𝔹κ∗γ⋆,t,sSkor] with length |γ|≥m+1 are all removed.

By (E.1), we see that ℰt,s2⁢φ⁢(x) has the form

for some ν=ν⁢(m)∈ℕ, multi-indices β⁢(ℓ) and bounded smooth functions C(ℓ):[0,T]×ℝN→ℝ with bounded derivatives, ℓ=1,…,ν. Note that (E.1) is equivalent to the following:

Then we have the assertion.

F Proof of Lemma 4.3

Since 𝒲(m)⁢(t,s,x,Bs-Bt) is given by the sum of the form

where v∈Cb∞⁢([0,T]×ℝN) and 𝒫r⁢(ξ) is polynomial of ξ=(ξ1,…,ξd) of degree r, there is C>0 such that

for all t<s and x∈ℝN, where we used the following property of the Brownian motion: for k=1,…,d,

G Proof of Lemma 4.4

By Lemma 4.2 and Lemma 4.3, we immediately have:

where C0,C1>0 are some constants independent of f:ℝN→ℝ, n≥1 and k≤n-1.

H Proof of Lemma 4.7

First, note that for r≥1, p=2⁢e, e∈ℕ and ℓ≤ν (=ν⁢(m)) there exists C⁢(T)>0 (independent of n) such that

for all x∈ℝN and k≤n, which can be checked in the following way. Let

Note that

is given by the sum of the form

where 𝒫r⁢(B(i-1)⁢T/n,i⁢T/n) is polynomial of degree r. Let {ℱt}t≥0 be the filtration generated by the Brownian motion. It holds that