MEAN-SQUARE EXPONENTIAL DICHOTOMY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

Hailong Zhu; Jifeng Chu

doi:10.11948/2016034

2016 Volume 6 Issue 2

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Hailong Zhu, Jifeng Chu. MEAN-SQUARE EXPONENTIAL DICHOTOMY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 463-478. doi: 10.11948/2016034

Citation:

Hailong Zhu, Jifeng Chu. MEAN-SQUARE EXPONENTIAL DICHOTOMY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 463-478. doi: 10.11948/2016034

MEAN-SQUARE EXPONENTIAL DICHOTOMY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

Hailong Zhu^1,2,
Jifeng Chu²

1 School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China;
2 Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

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Abstract

Abstract

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes.
- Mean-square exponential dichotomy /
- Euler-Maruyama method /
- stochastic differential equation
MSC: 34D09;65C30

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References

[1]	L. Arnold, Stochastic Differential Equations:theory and applications, New York, 1974. Google Scholar
[2]	L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, vol. 1926, Springer, 2008. Google Scholar
[3]	W. A. Coppel, Dichotomy in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York/Berlin, 1978. Google Scholar
[4]	J. M. DeLaurentis and B. A. Boughton, An asymptotic analysis of a generalized Langevin equation, Stochastic Process. Appl., 33(1989), 275-284. Google Scholar
[5]	L. C. Evans, An Introduction to Stochastic Differential Equations, Amer. Math. Soc., 2012. Google Scholar
[6]	A. Friedman, Stochastic differential equations and applications, Stochastic differential equations, 75-148, C.I.M.E. Summer Sch. 77, Springer, Heidelberg, 2010. Google Scholar
[7]	E. Hairer and G. Wanner, Solving Ordinary Differential Equations Ⅱ, Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin, 1996. Google Scholar
[8]	D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numerical Anal., 38(2000), 753-769. Google Scholar
[9]	D. J. Higham,An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43(2001), 525-546. Google Scholar
[10]	D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40(2002), 1041-1063 Google Scholar
[11]	D. J. Higham, X. Mao and A. M. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math., 6(2003), 297-313. Google Scholar
[12]	D. J. Higham, X. Mao and C. G. Yuan, Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations, Numer. Math., 108(2007), 295-325. Google Scholar
[13]	P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253(2012), 1422-1438. Google Scholar
[14]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. Google Scholar
[15]	X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. Google Scholar
[16]	J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, in:Pure and Applied Mathematics, vol. 21, Academic Press, 1966. Google Scholar
[17]	O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32(1930), 703-728. Google Scholar
[18]	Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33(1996), 2254-2267. Google Scholar
[19]	H. Schurz, Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise, Stochastic Anal. Appl., 14(1996), 313-354. Google Scholar
[20]	D. Stoica, Uniform exponential dichotomy of stochastic cocycles, Stochastic Process. Appl., 120(2010), 1920-1928. Google Scholar
[21]	D. Stoica and M. Megan, On nonuniform dichotomy for stochastic skewevolution semiflows in Hilbert spaces, Czechoslovak Math. J., 62(2012)(137), 879-887 Google Scholar