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Mean-square numerical approximations to random periodic solutions of stochastic differential equations
- Qingyi Zhan^{1, 2}Email author
Received: 7 April 2015
Accepted: 31 August 2015
Published: 17 September 2015
Abstract
This paper is devoted to the possibility of mean-square numerical approximations to random periodic solutions of dissipative stochastic differential equations. The existence and expression of random periodic solutions are established. We also prove that the random periodic solutions are mean-square uniformly asymptotically stable, which ensures the numerical approximations are feasible. The convergence of the numerical approximations by the random Romberg algorithm is also proved to be mean-square. A numerical example is presented to show the effectiveness of the proposed method.
Keywords
- stochastic differential equation
- random periodic solutions
- random Romberg algorithm
- pullback
- forward infinite horizon stochastic integral equations
1 Introduction
Stochastic differential equations (SDEs) have an important position in theory and application, for more details we refer the reader to [1] and [2]. In recent years, there has an increasing interest in random periodic solutions of SDEs. Random periodic solutions describe many physical phenomena and play an important role in aeronautics, electronics, biology, and so on [3, 4]. The existence of random periodic solutions was established by Feng et al. [5]. However, random periodic solutions have not been explicitly constructed as yet. Therefore, numerical approximations to random periodic solutions are an important method for studying their dynamic behavior. There are, however, few numerical studies in this field. The main difficulties lie in determining the initial value at the starting time and simulating improper integrals more efficiently. Therefore, we are concerned with the possibility of mean-square numerical approximation and numerical analysis of convergence in this paper.
There are two main motivations for this work. It is well known that, in the deterministic case, some researchers have obtained extensive results, including numerical approximations to periodic solution. We refer the reader to [6] and [7] and the references therein. However, few studies have been done in the random case. Yevik and Zhao [8] treated the numerical stationary solutions of SDEs. Liu et al. [3] investigated square-mean almost periodic solutions for a class of stochastic integro-differential equations. To the best of our knowledge, no investigations of mean-square numerical approximations to random periodic solutions of SDEs exist in the literature. Numerical approximation is still an interesting method for studying random periodicity in random dynamical systems.
Because there exist errors in the initial value at the starting time, and the random periodic solutions are sensitive to the initial value, we can only deal with SDEs whose random periodic solutions are mean-square uniformly asymptotically stable. The main results we obtain are the numerical approximations to random periodic solutions of dissipative SDEs, and the proof of mean-square convergence. This shows that mean-square numerical approximations to random periodic solutions are in fact close to the exact solutions and the iterative error can be controlled in the range of the presupposed error tolerance.
This paper is organized as follows. Section 2 deals with some preliminaries intended to clarify the presentation of concepts and norms used later. In Section 3 we present theoretical results on random periodic solutions of dissipative SDEs. This is the main conclusion of the article, which contains the existence and stability of random periodic solutions, the numerical implementation method and the mean-square convergence theorem. Section 4 is devoted to numerical experiments, which demonstrate that these algorithms can be applied to simulate random periodic solutions of dissipative SDEs. Finally, Section 5 gives some brief conclusions.
2 Preliminaries
Throughout the rest of this paper, we make the following notations.
For simplicity in notations, the norms \(\|\cdot\|_{2}\) and \(\| \cdot\| _{L^{2}(\Omega,P)}\) are usually written as \(\|\cdot\|\).
The following hypotheses are made for the theoretical analysis.
Hypothesis 2.1
- (i)
There exists a constant \(K^{*}>0\) such that \(\|x_{0}\|\leq K^{*}\).
- (ii)The mapping \(f:R\times R^{d}\rightarrow R^{d} \) is continuous, and there exist positive constants \(J_{1}\) and \(K_{1}\) such that \(f(t,0)\) is globally bounded with \(|f(t,0)|\leq J_{1}\) and for any \(X_{1},X_{2}\in R^{d}\), the following inequality holds:$$ \bigl|f(t,X_{1})-f(t,X_{2})\bigr|\leq K_{1}|X_{1}-X_{2}|. $$(6)
- (iii)
The mapping \(g:R\rightarrow R^{d} \) is continuous, and there exists a positive constant \(J_{2}\) such that \(g(t)\) is globally bounded with \(|g(t)|\leq J_{2}\).
3 Theoretical results
3.1 Existence of random periodic solutions
The following result guarantees the existence of random periodic solutions for dissipative SDEs, and is a direct consequence of Theorem 3.2.4 in [4].
Lemma 3.1
- (i)
\(\varphi(s,t,\omega)\cdot:B\rightarrow B\) is a.s. continuous;
- (ii)
\(\varphi(s+\tau,t+\tau,\omega)x_{0}=\varphi (s,t,\theta_{\tau}\omega)x_{0}\);
- (iii)
there exist constants \(c\in(0,1)\) and \(M>0\) such that \(\|\varphi(s,t,\omega)x_{0}-\varphi(s,t,\omega)\hat{x}_{0}\|\leq c^{t-s}\| x_{0}-\hat{x}_{0}\|\) and \(\|\varphi(s,t,\omega)x_{0}\|\leq M\), where M may depend on \((t-s)\),
Lemma 3.2
Proof
So it is valid for the one-dimensional case.
From the conclusions of Lemmas 3.1 and 3.2, we obtain the following theorem.
Theorem 3.3
Proof
In order to utilize Lemma 3.1 to this problem, we only need to check that the conditions of this theorem satisfy its three hypotheses.
First and foremost, by the assumptions of SDE (1), the hypothesis (i) obviously holds.
This completes the check of the third hypothesis.
3.2 Stability
In this section we investigate the mean-square uniformly asymptotic stability of the random periodic solution \(Y(t,\omega)\) of SDE (1). The pullback method is a powerful tool in the proof of uniformly asymptotic stability. To be precise, let us introduce some related definitions [10].
Definition 3.1
(ii) The random periodic solution \(Y(t,\omega)\) of SDE (1) is said to be mean-square uniformly stable if for any given \(\epsilon>0\) and every other random periodic solution \(\hat{Y}(t,\omega )\) of SDE (1), there exists \(\delta=\delta(\epsilon)\) such that \(\|x_{0}-\hat{x}_{0}\|\leq\delta\) implies the inequality \(\|Y(t,\omega )-\hat{Y}(t,\omega)\|<\epsilon\) holds for any \(t\geq s\), where \(s=t-m\tau\).
(iii) The random periodic solution \(Y(t,\omega)\) of SDE (1) is said to be mean-square uniformly asymptotically stable if it is mean-square uniformly stable and mean-square asymptotically stable.
Theorem 3.4
Assume that for any initial values \(x_{0} \) and \(\hat{x}_{0}\in L^{2}(\Omega,P)\), the coefficients of SDE (1) satisfy Theorem 3.3, then the random periodic solution \(Y(t,\omega)\) of SDE (1) is mean-square uniformly asymptotically stable.
Proof
Then by Definition 3.1(i), it is mean-square asymptotically stable.
Secondly, let \(V(s,t,\omega)\bar{x}_{0}=Y(t,\omega)-\hat{Y}(t,\omega)\), where \(\bar{x}_{0}=(x_{0},\hat{x}_{0})\). It is the fact that \(V(s,t,\omega )\bar{x}_{0}\) is also a random periodic solution of SDE (1). Without loss of generality, we only consider the case \(s\geq0\). The proof of other case is similar, that is, by the transformation \(\breve {s}=s+m'\tau\), we can change the case of \(s\leq0\) to the case of \(\breve{s}\geq0\), where \(m'\) is a positive integer. Let \(\bar{x}'_{0}\) be the initial value at the starting time \(s=0\). From the above result, that is, the mean-square asymptotic stability, it follows that for any given \(\epsilon>0\), there exists \(\delta_{0}=\delta_{0}(\epsilon )>0\) such that \(\|x'_{0}-\hat{x}'_{0}\|\leq\delta_{0}\) implies the inequality \(\|V(0,t,\omega)\bar{x}'_{0}\|<\epsilon\) holds for \(t\geq0\).
For the first case \(s\in[0,\tau]\), by the fact that \(V(s,t,\omega)\bar {x}_{0}\) is continuous with respect to \((s,\bar{x}_{0})\) and uniformly continuous with respect to s for \(s\in[0,\tau]\), there exists \(\delta=\delta(\epsilon)>0\) such that \(\|x_{0}-\hat{x}_{0}\| \leq\delta\) implies the inequality \(\|V(s,0,\omega)\bar{x}_{0}\|<\delta _{0} \) holds for \(s\in[0,\tau]\).
Then \(\|x_{0}-\hat{x}_{0}\|\leq\delta\) implies the inequality \(\|Y(t,\omega )-\hat{Y}(t,\omega)\|<\epsilon\) holds for any \(s\geq0\) and \(t\geq s\).
Therefore it follows from Definition 3.1(ii) that it is mean-square uniformly asymptotically stable. The conclusion follows from Definition 3.1(iii). This completes the proof. □
3.3 Numerical implementation method of random periodic solutions
It follows from Theorem 3.3 that the forward infinite horizon integral equation (9) is the random periodic solution of SDE (1). However, if the numerical method is applied to the improper integral (9), only the numerical approximations to (9) are obtained. Therefore, approximating the random periodic solution requires that the random periodic solution (9) is mean-square uniformly asymptotically stable. It follows from Theorem 3.4 that the random periodic solution (9) is mean-square uniformly asymptotically stable. This implies that the numerical solution to initial problem is the numerical solution to the random periodic solution.
By means of reselecting the corresponding starting time and \(s'\), we can simulate a random periodic solution in an arbitrary finite time interval with any given presupposed error tolerance.
In order to improve the accuracy of the integral, the random Romberg algorithm is applied to (18) and \(\bar{Y}(0,\omega)\). The method applied to (18) in detail is shown as follows.
The method applied to \(\bar{Y}(0,\omega)\) in detail is shown as follows, which is similar to the former.
3.4 Convergence
Theorem 3.5
Assume that for any initial value \(x_{0}\in L^{2}(\Omega,P)\), the coefficients of SDE (1) satisfy Theorems 3.3 and 3.4, then the numerical approximation \(\tilde{Y}_{N}(t,\omega)\) to random periodic solutions of SDE (1) by the random Romberg algorithm is mean-square convergent.
Proof
4 Numerical experiments
Similarly, we can choose another presupposed initial error tolerance \(\delta=0.011\) such that \(s'=-35\) and \(x_{0}=0.15\) are determined, which satisfy Theorem 3.3, too.
5 Conclusion
Finally, conclusions and future work are summarized. In this paper, the possibility of mean-square numerical approximations to random periodic solutions of SDEs is discussed. The random Romberg algorithm is shown in detail. The results show that the method is effective and universal; numerical experiments are performed and match the results of theoretical analysis. In our further work, we will consider more simple and practical methods which will be used to simulate a broader class of SDEs whose diffusion coefficient is a function of t and x.
Declarations
Acknowledgements
The author would like to express his gratitude to Prof. Jialin Hong for his helpful discussion. This work is supported by NSFC (Nos. 11021101, 11290142, and 91130003).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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