Abstract
This paper proposes a new Markov chain approach to second-order weak approximations of stochastic differential equations (SDEs) driven by d-dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order, and any discrete moment-matched random variables or the Lévy area simulation method are not used. The required number of random variables is still d in one-step simulation of the implementation of the scheme. In the Markov chain, a correction term with Lie bracket of vector fields associated with SDEs appears as the cost of not using moment-matched random variables.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 16K13773
Funding statement: This work is supported by JSPS KAKENHI (Grant Number 16K13773) from MEXT, Japan and a research fund from Tokio Marine Kagami Memorial Foundation.
A Appendix
Before we show the proofs of Lemma 2.1, Proposition 2.2, Theorem 2.4 and Corollary 2.6, we prepare some notations and results on stochastic calculus.
A.1 Stochastic calculus
We summarize the analysis of Wiener functionals, which will be a key tool for the proofs.
Please see [17, 13] for the details.
Let
where
or
The iterated derivatives can be defined as
For
The adjoint operator
In particular, for
Here we note that the stochastic integral on the right-hand side is the Itô integral, and also
holds by the chain rule of the Malliavin derivative.
Define the space of smooth Wiener functionals
is invertible a.s. and
where
Here,
Recall that the space of Watanabe distributions
Furthermore,
In particular, let
See, for example, [8, 14, 15, 19, 21, 20] for the details on computations for Wiener functionals and applications.
A.2 Key lemma
In the proofs of Lemma 2.1, Proposition 2.2 and Theorem 2.4, the following two lemmas will play an important role.
Lemma A.1.
Let h be a bounded adapted process. Then, for
for all
Lemma A.2.
Let
for all
Proof.
Let
Also, if
Define the multi-index
where
it holds
Therefore we have
for some
Lemma A.2 is a slight generalization of [20, Lemma 1].
In the following, we typically take the function h as
A.3 Proof of Lemma 2.1
Let
where
We expand
(A.6)where
We immediately obtain
since, for
Here, we used Lemma A.1, in particular, for
To obtain
for
Using Lemma A.2, for
Then we get
We use Lemma A.2 again to attain
We also remark that the product of iterated Itô integrals in the third term on the right-hand side of (A.6) is represented through the Itô formula as
(A.7)for
where
(A.8)where
We finally compute the second and third term on the right-hand side of (A.8). Using equation (A.3), for
By the integration by parts for Watanabe distributions (A.4), with the computation (A.5), we have
and get
We obviously have
Therefore, we obtain the assertion. ∎
A.4 Proof of Proposition 2.2
Let
and expand
Then we get
(A.9)where
Then we can immediately observe that
(A.10)Applying formula (A.7) and Lemma A.2 to (A.10), we have the following expansion for the third term on the right-hand side of (A.9):
where
Then we have
(A.11)where
To obtain the same expansion in Lemma 2.1, we need to adjust the third and the fourth terms on the right-hand side of (A.11) by adding some correction terms to
(A.12)Here, the Malliavin weights
(A.13)We will use the relation
in the following. The expectation
(A.14)where
with
From the definition (A.12) of
Then we have
Also, since
holds by (A.13), we have
Then we obtain
In order to show the bound of
considering (A.13). Note that it holds
By applying Lemma A.2, we have
Also, for
Then we obtain
and therefore the error
The final and crucial step is as follows. We apply the integration by parts (A.2) to the third term on the right-hand side of (A.14) as
to get the following representation for
A.5 Proof of Theorem 2.4
The one-step small time approximation in Theorem 2.3 is summarized as follows: there is a
(A.15)In the proof, the generic constant
We easily see that
In order to get the global approximation, it suffices to show the bound of
(A.16)by Kusuoka and Stroock [6]. By (A.15) and (A.16), we have
For
using a similar argument as in [4, 19]. When
We finally show the bound of
Then the following small time approximations around
for
A.6 Proof of Corollary 2.6
When
where
for
for all
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