Filtering and 1/3 Power Law for Optimal Time Discretisation in Numerical Integration of Stochastic Differential Equations

Abstract: This paper is concerned with the numerical integration of stochastic differential equations (SDEs) which govern diffusion processes driven by a standard Wiener process. With the latter being replaced by a sequence of increments at discrete moments of time, we revisit a filtering point of view on the approximate strong solution of the SDE as an estimate of the hidden system state whose conditional probability distribution is updated using a Bayesian approach and Brownian bridges over the intermediate time intervals. For a class of multivariable linear SDEs, where the numerical solution is organised as a Kalman filter, we investigate the fine-grid asymptotic behaviour of terminal and integral mean-square error functionals when the time discretisation is specified by a sufficiently smooth monotonic transformation of a uniform grid. This leads to constrained optimisation problems over the time discretisation profile, and their solutions reveal a 1/3 power law for the asymptotically optimal grid density functions. As a one-dimensional example, the results are illustrated for the Ornstein-Uhlenbeck process.