Stationary in Distributions of Numerical Solutions for Stochastic Partial Differential Equations with Markovian Switching

Abstract and Applied Analysis

Research Article

Yan Li¹  Yi Shen² 

[1] Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China, hust.edu.cn;College of Science, Huazhong Agriculture University, Wuhan 430079, China, hzau.edu.cn;Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China, hust.edu.cn

Others  :  1297427
DOI  :  10.1155/2013/752953

 received in 2012-12-30, accepted in 2013-02-24,  发布年份 2013

【 摘 要 】

We investigate a class of stochastic partial differential equations with Markovian switching. By using the Euler-Maruyama scheme both in time and in space of mild solutions, we derive sufficient conditions for the existence and uniqueness of the stationary distributions of numerical solutions. Finally, one example is given to illustrate the theory.

【 授权许可】

CC BY   
Copyright © 2013 Yi Shen and Yan Li. 2013

【 参考文献 】

• [1]G. Da Prato, J. Zabczyk. (1992). Stochastic Equations in Infinite-Dimensions. DOI: 10.1017/CBO9780511666223.
• [2]A. Debussche. (2011). Weak approximation of stochastic partial differential equations: the nonlinear case. Mathematics of Computation.80(273):89-117. DOI: 10.1017/CBO9780511666223.
• [3]I. Gyöngy, A. Millet. (2009). Rate of convergence of space time approximations for stochastic evolution equations. Potential Analysis.30(1):29-64. DOI: 10.1017/CBO9780511666223.
• [4]A. Jentzen, P. E. Kloeden. (2011). Taylor Approximations for Stochastic Partial Differential Equations. DOI: 10.1017/CBO9780511666223.
• [5]T. Shardlow. (1999). Numerical methods for stochastic parabolic PDEs. Numerical Functional Analysis and Optimization.20(1-2):121-145. DOI: 10.1017/CBO9780511666223.
• [6]M. Athans. (1987). Command and control (C2) theory: a challenge to control science. IEEE Transactions on Automatic Control.32(4):286-293. DOI: 10.1017/CBO9780511666223.
• [7]D. J. Higham, X. Mao, C. Yuan. (2007). Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations. Numerische Mathematik.108(2):295-325. DOI: 10.1017/CBO9780511666223.
• [8]X. Mao, C. Yuan. (2006). Stochastic Differential Equations with Markovian Switching. DOI: 10.1017/CBO9780511666223.
• [9]M. Mariton. (1990). Jump Linear Systems in Automatic Control. DOI: 10.1017/CBO9780511666223.
• [10]J. Bao, Z. Hou, C. Yuan. (2010). Stability in distribution of mild solutions to stochastic partial differential equations. Proceedings of the American Mathematical Society.138(6):2169-2180. DOI: 10.1017/CBO9780511666223.
• [11]J. Bao, C. Yuan. Numerical approximation of stationary distribution for SPDEs. . DOI: 10.1017/CBO9780511666223.
• [12]X. Mao, C. Yuan, G. Yin. (2005). Numerical method for stationary distribution of stochastic differential equations with Markovian switching. Journal of Computational and Applied Mathematics.174(1):1-27. DOI: 10.1017/CBO9780511666223.
• [13]C. Yuan, X. Mao. (2005). Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence. Journal of Difference Equations and Applications.11(1):29-48. DOI: 10.1017/CBO9780511666223.
• [14]C. Yuan, X. Mao. (2003). Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stochastic Processes and their Applications.103(2):277-291. DOI: 10.1017/CBO9780511666223.
• [15]W. J. Anderson. (1991). Continuous-Time Markov Chains. DOI: 10.1017/CBO9780511666223.