Numerical solutions of stochastic functional differential equations

In this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

[1]  Andrew M. Stuart,et al.  Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations , 2002, SIAM J. Numer. Anal..

[2]  S. Mohammed.  Stochastic functional differential equations , 1984 .

[3]  X. Mao,et al.  Numerical solutions of stochastic differential delay equations under local Lipschitz condition , 2003 .

[4]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[5]  Xuerong Mao,et al.  Approximate solutions for a class of stochastic evolution equations with variable delays. II , 1991 .

[6]  Evelyn Buckwar,et al.  NUMERICAL ANALYSIS OF EXPLICIT ONE-STEP METHODS FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS , 1975 .

[7]  E. Helfand.  Numerical integration of stochastic differential equations , 1979, The Bell System Technical Journal.

[8]  I. Gyöngy,et al.  Existence of strong solutions for Itô's stochastic equations via approximations , 1996 .

[9]  Evelyn Buckwar,et al.  Introduction to the numerical analysis of stochastic delay differential equations , 2000 .

[10]  E. Platen.  An introduction to numerical methods for stochastic differential equations , 1999, Acta Numerica.

[11]  I. Gyöngy.  A note on Euler's Approximations , 1998 .

[12]  István Gyöngy,et al.  A Note on Euler's Approximations , 1998, Potential Analysis.

[13]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[14]  E. Platen,et al.  Strong discrete time approximation of stochastic differential equations with time delay , 2000 .

[15]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[16]  R. Seydel.  Numerical Integration of Stochastic Differential Equations , 2004 .

[17]  Yaozhong Hu.  Semi-Implicit Euler-Maruyama Scheme for Stiff Stochastic Equations , 1996 .

[18]  Pierre Bernard,et al.  Convergence of Numerical Schemes for Stochastic Differential Equations , 2001, Monte Carlo Methods Appl..