Implicit Taylor methods for stiff stochastic differential equations

In this paper we discuss implicit Taylor methods for stiff Ito stochastic differential equations. Based on the relationship between Ito stochastic integrals and backward stochastic integrals, we introduce three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0 and the implicit Taylor method with strong order 1.5. The mean-square stability properties of the implicit Euler-Taylor and Milstein-Taylor methods are much better than those of the corresponding semi-implicit Euler and Milstein methods and these two implicit methods can be used to solve stochastic differential equations which are stiff in both the deterministic and the stochastic components. Numerical results are reported to show the convergence properties and the stability properties of these three implicit Taylor methods. The stability analysis and numerical results show that the implicit Euler-Taylor and Milstein-Taylor methods are very promising methods for stiff stochastic differential equations.

[1]  Peter E. Kloeden,et al.  A survey of numerical methods for stochastic differential equations , 1989 .

[2]  E. Platen,et al.  Balanced Implicit Methods for Stiff Stochastic Systems , 1998 .

[3]  Tianhai Tian,et al.  A note on the stability properties of the Euler methods for solving stochastic differential equations , 2000 .

[4]  Hans Christian Öttinger,et al.  Stochastic Processes in Polymeric Fluids , 1996 .

[5]  Zeev Schuss,et al.  Theory and Applications of Stochastic Differential Equations , 1980 .

[6]  Kevin Burrage,et al.  A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations , 1997 .

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

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

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

[10]  W. P. Petersen,et al.  A General Implicit Splitting for Stabilizing Numerical Simulations of Itô Stochastic Differential Equations , 1998 .

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

[12]  K. Burrage,et al.  High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations , 1996 .

[13]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[14]  Yoshihiro Saito,et al.  Stability Analysis of Numerical Schemes for Stochastic Differential Equations , 1996 .

[15]  Peter E. Kloeden,et al.  Stratonovich and Ito Stochastic Taylor Expansions , 1991 .

[16]  Tianhai Tian,et al.  The composite Euler method for stiff stochastic differential equations , 2001 .

[17]  Eckhard Platen,et al.  Simulation studies on time discrete diffusion approximations , 1987 .

[18]  Tianhai Tian,et al.  Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations , 2002 .

[19]  Kevin Burrage,et al.  General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems , 1998 .

[20]  S. S. Artemiev,et al.  Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations , 1997 .

[21]  Eckhard Platen,et al.  On weak implicit and predictor-corrector methods , 1995 .