Simulation of elliptic and hypo-elliptic conditional diffusions

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.