Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's

Abstract A general weak limit theorem for solutions of stochastic differential equations driven by arbitrary semimartingales is applied to give a unified treatment of limit theorems for random evolutions and consistency results for numerical schemes for stochastic differential equations. The asymptotic distribution of the error in an Euler scheme is studied. The Wong-Zakai correction in the random evolution limit arises through an integration by parts.

[1]  G. N. Mil’shtejn.  Approximate Integration of Stochastic Differential Equations , 1975 .

[2]  H. Rootzén.  Limit Distributions for the Error in Approximations of Stochastic Integrals , 1980 .

[3]  R. Khas'minskii.  A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .

[4]  W. Wagner,et al.  Undiased monte carlo estimators for functionals of weak solutions of stochastic diffretial equations , 1989 .

[5]  D. Talay,et al.  Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .

[6]  R. Hersh,et al.  Random evolutions, markov chains, and systems of partial differential equations. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[7]  G. Papanicolaou,et al.  Non-commuting random evolutions, and an operator-valued Feynman-Kac formula , 1972 .

[8]  W. Rüemelin.  Numerical Treatment of Stochastic Differential Equations , 1982 .

[9]  L. Slominski.  Stability of strong solutions of stochastic differential equations , 1989 .

[10]  J. Mémin,et al.  Condition UT et stabilité en loi des solutions d’équations différentielles stochastiques , 1991 .

[11]  Thomas G. Kurtz,et al.  A limit theorem for perturbed operator semigroups with applications to random evolutions , 1973 .

[12]  P. Protter,et al.  Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations , 1991 .

[13]  D. Talay,et al.  Discretization and simulation of stochastic differential equations , 1985 .

[14]  J. Mémin,et al.  Convergence en loi des suites d'intégrales stochastiques sur l'espace $$\mathbb{D}$$ 1 de Skorokhod , 1989 .

[15]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .