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

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.