{"ID":2886160,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.03135","arxiv_id":"2508.03135","title":"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.","short_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...","url_abs":"https://arxiv.org/abs/2508.03135","url_pdf":"https://arxiv.org/pdf/2508.03135v1","authors":"[\"Igor G. Vladimirov\"]","published":"2025-08-05T06:31:50Z","proceeding":"eess.SY","tasks":"[\"eess.SY\",\"math.OC\",\"math.PR\"]","methods":"[\"Diffusion Model\",\"Generative Adversarial Network\"]","has_code":false}