Adaptive stepsize implementations of implicit stochastic Runge Kutta methods with modified Wiener increment

Abstract

An adaptive stepsize algorithm is implemented on a stochastic implicit strong order 1 method, namely a stiffly accurate diagonal implicit stochastic Runge-Kutta method where a modified Wiener increment [?(?W n )] is involved instead of a regular ?W n = ??h, ??N(0,1) to avoid unboundedness. The modified Wiener increment is equal to the regular one only if ??? ? A h , otherwise [?(?W)] = ?A h ?h or A h ?h if ? ? ?A h or ? ? A h , respectively. The parameter A h is determined based on a strong order inequality requirement (Milstein et al., 2002) ?? The variable stepsize algorithm is based on Richardson's extrapolation. For every step change additional work is required to compute [?(?W n )] so that the correct Brownian path is maintained from simulation to simulation. Numerical experiments in solving nonlinear and linear stochastic differential equation problems demonstrate that better approximations are obtained with the modified Wiener increment compared to using the regular Wiener increment.