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

TL;DR

This paper introduces adaptive variable step-size schemes for weak solutions of SDEs, controlling conditional moments without sampling, to improve stability and accuracy in computing diffusion functionals.

Contribution

It develops a novel methodology for designing adaptive weak schemes for SDEs based on controlling local discrepancies of conditional moments, avoiding sampling.

Findings

Adaptive schemes improve stability over fixed step-size methods.

Numerical results demonstrate enhanced accuracy in computing diffusion functionals.

Proposed methods effectively handle instability issues in SDE simulations.

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 match 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 a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.