Simulation of Conditioned Semimartingales on Riemannian Manifolds

We present a scheme for simulating conditioned semimartingales taking values in Riemannian manifolds. Extending the guided bridge proposal approach used for simulating Euclidean bridges, the scheme replaces the drift of the conditioned process with an approximation in terms of a scaled radial vector field. This handles the fact that transition densities are generally intractable on geometric spaces. We prove the validity of the scheme by a change of measure argument, and we show how the resulting guided processes can be used in importance sampling and for approximating the density of the unconditioned process. The bridge sampling is experimentally illustrated on two- and three-dimensional manifolds. Here, we compare density estimates using the sampling scheme to approximations using heat kernel expansions, and we use the scheme to estimate the diffusion mean of sampled data.