Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

Abstract

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the matching between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce three variable step-size Euler schemes derived from three different discrepancy functions, as well as four variable step-size higher order weak schemes are proposed. Third, to compute the expectation of functionals of diffusion processes a general procedure for designing adaptive schemes with variable step-size and sample-size is presented, which combines a conventional Monte Carlo technique for estimating the total number of simulations with the new variable step-size weak schemes. Finally, a variety of numerical simulations are presented to show the potential of the introduced variable stepsize strategies and adaptive schemes to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.