Simulation of elliptic and hypo-elliptic conditional diffusions

TL;DR

This paper develops a unified method for simulating conditioned multidimensional diffusions, including hypo-elliptic cases, based on guided proposals, with applications to complex biological models.

Contribution

It extends the use of guided proposals to both elliptic and hypo-elliptic diffusions, even when observations are partial and linear.

Findings

Effective sampling in challenging diffusion models

Successful application to biological systems

Unified approach for different diffusion types

Abstract

Suppose $X$ is a multidimensional diffusion process. Assume that at time zero the state of $X$ is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix $L$. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in Schauer et al. (2017), can be used in a unified way for both uniformly and hypo-elliptic diffusions, also when $L$ is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.