Strong convergence rate of principle of averaging for jump-diffusion processes

Abstract

We study jump-diffusion processes with two well-separated time scales. It is proved that the rate of strong convergence to the averaged effective dynamics is of order O(ϵ^1/2), where ϵ\ll 1 is the parameter measuring the disparity of the time scales in the system. The convergence rate is shown to be optimal through examples. The result sheds light on the designing of efficient numerical methods for multiscale stochastic dynamics.

Keywords

Stochastic differential equation / time scale separation / averaging of perturbations

References

[1] E W, Engquist B. The heterogeneous multiscale methods. Commun Math Sci, 2003, 1(1): 87-133

[2] E W, Liu D, Vanden-Eijnden E. Analysis of multiscale methods for stochastic differential equations. Commun Pure Appl Math, 2005, 58(11): 1544-1585

[3] Freidlin M I, Wentzell A D. Random Perturbations of Dynamical Systems. 2nd ed. New York: Springer-Verlag, 1998

[4] Givon D. Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. SIAM Mul Mod Simu, 2007, 6: 577-594

[5] Givon D, Kevrekidis I G, Kupferman R. Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems. Commun Math Sci, 2006, 4(4): 707-729

[6] Khasminskii R Z. Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion. Theory Probab Appl, 1963, 8: 1-21

[7] Khasminskii R Z. Stochastic Stability of Differential Equations. Alphen aan den Rijn: Sijthoff & Noordhoff, 1980

[8] Khasminskii R Z, Yin G. On averaging principles: An asymptotic expansion approach. SIAM J Math Anal, 2004, 35(6): 1534-1560, 2004

[9] Kifer Y. Stochastic versions of Anosov and Neistadt theorems on averaging. Stoch Dyn, 2001, 1(1): 1-21

[10] Kushner H J. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, 3, Systems & Control: Foundations & Applications. Boston: Birkhäuser, 1990

[11] Liu D. Strong convergence of principle of averaging for multiscale stochastic dynamical systems. Commun Math Sci, 2010, 8: 999-1020

[12] Liu D. Analysis of multiscale methods for stochastic dynamical systems with multiple time scales. SIAM Mul Mod Simu, 2010, 8: 944-964

[13] Menaldi J L, Robin M. Invariant measure for diffusions with jumps. Appl Math Optim, 1999, 40: 105-140

[14] Meyn S P, Tweedie R L. Stability of Markovian processes, I. Adv Appl Probab, 1992, 24(3): 542-574

[15] Meyn S P, Tweedie R L. Stability of Markovian processes, II. Adv Appl Probab, 1993, 25(3): 487-517

[16] Meyn S P, Tweedie R L. Stability of Markovian processes, III. Adv Appl Probab, 1993, 25(3): 518-548

[17] Vanden-Eijnden E. Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun Math Sci, 2003, 1: 377-384

[18] Veretennikov A Yu. On an averaging principle for systems of stochastic differential equations. Math Sbornik, 1990, 181(2): 256-268

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg