Improved bridge constructs for stochastic differential equations

TL;DR

This paper introduces novel methods for efficiently generating diffusion bridges in nonlinear stochastic differential equations, especially under partial and noisy observations, by partitioning the process and using linear residual approximations.

Contribution

It proposes new bridge constructs that improve sampling efficiency for nonlinear SDEs with partial, noisy data, extending existing methods with linear residual approximations.

Findings

New bridge constructs outperform existing methods in simulations.

Partitioning the process improves sampling efficiency.

Approaches are validated on three different applications.

Abstract

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an It\^o stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler-Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.