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.1 Proof of the probability representation of (C1)
(see e.g. ), and the fundamental solution is known to be the probability density function of the Wiener process on reflected at 0. Therefore, we obtain the representation
where is an RV with the standard normal distribution. Moreover, the representation of follows from standard computations.
For any , there exist
an RV 𝛽 with distribution,
an RV Γ with distribution,
an RV with standard normal distribution
Since the fundamental solution is given by (A.1), we have
where and are defined by
and and the Wiener process 𝑊 are mutually independent. Then we have
Let be a distributed random variable which admits the probability density function
and such that and the Wiener process 𝑊 are mutually independent.
A.2 Proof of the probability representation of (C2)
Then we have the following representation:
where is an RV with standard normal distribution. Moreover, from an analogue of Lemma A.1, we obtain
A.3 Proof of the probability representation of (C3)
We now define the integer-valued functions and ,
Then we obtain the following representation for .
For any , it holds that
where is an RV with standard normal distribution.
The proof of Lemma A.2 follows from standard computation, and therefore we omit it.
In order to study the representation of , we first consider the square of the fundamental solution (A.2). Note that is an even function and the linear map is in one-to-one correspondence for each ; thus we obtain
where and are defined by
By using the importance sampling technique and (A.3), we obtain the following representation for .
For any , there exist RVs and for each such that are mutually independent and
where and are Kronecker deltas, and and are defined by
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 , we split the first term of (A.6) as follows:
Note that for any and ; thus we have
where and are defined by
In the same way,
where and are defined by
Therefore, we have concluded the proof. ∎
A.4 Proof of the probability representation of (C4)
Hence we obtain the following lemma using the same arguments as in the proof of Lemma A.3.
For any , there exist RVs and for each such that are mutually independent, respectively, and
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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