Abstract
The paper proposes a new second-order discretization method for forward-backward stochastic differential equations. The method is given by an algorithm with polynomials of Brownian motions where the local approximations using Malliavin calculus play a role. For the implementation, we introduce a new least squares Monte Carlo method for the scheme. A numerical example is illustrated to check the effectiveness.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 19K13736
Funding statement: This work is supported by JSPS KAKENHI (Grant Number 19K13736) from MEXT, Japan.
A Basics of Malliavin calculus and useful lemmas
We briefly summarize Malliavin calculus which is necessary for the main section and the discussion below.
See [7, 11] for details of Malliavin calculus.
On the probability space
where
For
Moreover, we write
We say that
For
and
The integration by parts on Wiener space is given as follows.
Proposition A.1.
Let
and
We define the space of Watanabe distributions
For more details on computation with Watanabe distributions, see [12], for example.
Hereafter, let
Here, the notation
We list the following technical lemmas (Lemmas A.2–A.7) which are used in Appendix B.
Lemma A.2.
Proof.
See [10], for example. ∎
Lemma A.3.
Proof.
See [10], for example. ∎
Lemma A.4.
Let
Let
for a multi-index β satisfying
Proof.
Apply the duality formula.∎
Lemma A.5.
Let
for all
Proof.
See [10], for example. ∎
Lemma A.6.
Proof.
See [10], for example. ∎
Lemma A.7.
The following formulas hold:
Proof.
See [10], for example. ∎
B Local approximations
The following local approximations (Theorems B.1 and B.2) with
Theorem B.1.
For
for all
Proof.
See [10, Proposition 3.9]. ∎
The expectation with Crisan–Manolarakis’s
Theorem B.2.
For
for all
Proof.
By the expansion of
with the residual
where
with
Some terms in (B.1) will be
for some
by the Cauchy–Schwarz inequality with
Also, for
For
for a multi-index β satisfying
Also, using a similar computation in (B.2), we can show that, for
Therefore, we have
for some
We further analyze the approximation terms. Using Lemma A.2 and Lemma A.3, we have
Here, the terms corresponding to the case
Applying the integration by parts formulas in Lemma A.7, we get
C Proof of the main theorem (Theorem 2.4)
We define the following:
for
where
for
by [4]. Then, in order to estimate (C.2), it suffices to show the bound of
The following lemma finishes the proof of Theorem 2.4.
Lemma C.1.
There exists
for all
Proof.
We have
Here,
with
and
with
are recursively defined.
For
Note that we have
In the following, we use a generic constant
For the last term in the right-hand side of the above, we have
by (C.4) and the estimate
Inductively, we have
Then the estimate
is reduced to that of
and
We note that
Applying Theorem B.1, we get
By (C.1) and (C.3) with Theorem B.2, it holds that
Then, by the estimate
and by the fact
Therefore, for
With a similar argument, we also have
Finally, we get
D Some useful results
When
where the divergence formula (2.3) is applied.
Then we can directly obtain the integration by parts using the inverse of
Corollary D.1.
When
Corollary D.2.
When
Remark D.3.
Here,
We also have
Furthermore, we have
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