S-rock Methods for Stiff Itô Sdes

In this paper, we present a class of explicit numerical methods for stiff Itô stochastic differential equations (SDEs). These methods are as simple to program and to use as the well-known Euler-Maruyama method, but much more efficient for stiff SDEs. For such problems, it is well known that standard explicit methods face step-size reduction. While semi-implicit methods can avoid these problems at the cost of solving (possibly large) nonlinear systems, we show that the stepsize reduction phenomena can be reduced significantly for explicit methods by using stabilization techniques. Stabilized explicit numerical methods called S-ROCK (for stochastic orthogonal RungeKutta Chebyshev) have been introduced in [C. R. Acad. Sci. Paris, vol. 345, no. 10, 2007] as an alternative to (semi-) implicit methods for the solution of stiff stochastic systems. In this paper we discuss a genuine Itô version of the S-ROCK methods which avoid the use of transformation formulas from Stratonovich to Itô calculus. This is important for many applications. We present two families of methods for one-dimensional and multi-dimensional Wiener processes. We show that for stiff problems, significant improvement over classical explicit methods can be obtained. Convergence and stability properties of the methods are discussed and numerical examples as well as applications to the simulation of stiff chemical Langevin equations are presented.