Abstract
The aim of this paper is to study the simulation of the expectation for the solution of linear stochastic partial differential equation driven by the space-time white noise with the bounded measurable coefficient and different boundary conditions. We first propose a Monte Carlo type method for the expectation of the solution of a linear stochastic partial differential equation and prove an upper bound for its weak rate error. In addition, we prove the central limit theorem for the proposed method in order to obtain confidence intervals for it. As an application, the Monte Carlo scheme applies to the stochastic heat equation with various boundary conditions, and we provide the result of numerical experiments which confirm the theoretical results in this paper.
A Appendix
A.1 Proof of the probability representation of (C1)
We consider the solution of SPDE (3.1) with condition (C1).
It is well known that the fundamental solution of PDE (2.1) with
(see e.g. [19]), and the fundamental solution is known to be the probability density function of the Wiener process on
where
For any
an RV 𝛽 with
an RV Γ with
an RV
Proof
Since the fundamental solution
where
and
Let
and such that
A.2 Proof of the probability representation of (C2)
We consider the solution of SPDE (3.1) with condition (C2).
It is well known that the fundamental solution of PDE (2.1) with
Then we have the following representation:
where
A.3 Proof of the probability representation of (C3)
We consider the solution of SPDE (3.1) under condition (C3).
It is well known that the fundamental solution of PDE (2.1) with
for any
We now define the integer-valued functions
Then we obtain the following representation for
For any
where
The proof of Lemma A.2 follows from standard computation, and therefore we omit it.
In order to study the representation of
where
By using the importance sampling technique and (A.3), we obtain the following representation for
For any
where
Proof
Using (A.3), we obtain
We now consider the first term of (A.6).
In order to derive a probability representation that satisfies (2.6) for
Note that
where
where
In the same way,
where
Therefore, we have concluded the proof. ∎
A.4 Proof of the probability representation of (C4)
We consider the solution of SPDE (3.1) with condition (C4).
It is well known that the fundamental solution of PDE (2.1) with
(see e.g. [19, p. 52]).
Using the same arguments as in Subsection A.3, we have the following representation for
Moreover, by using
Hence we obtain the following lemma using the same arguments as in the proof of Lemma A.3.
For any
Acknowledgements
The authors would like to thank Professor Arturo Kohatsu-Higa for his valuable suggestions and fruitful discussions. The authors would also like to thank Associate Professor Dai Taguchi for his careful reading and advice. The authors would also like to thank an anonymous referee for his/her careful readings.
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