Bridge change-of-measure for propagator estimation of diffusions

Abstract

The propagator of one-dimensional diffusions has relevant applications in statistical inference and across many disciplines. Yet it is obtained in closed form only for very few simple models. Here we analyze an importance sampling strategy based on a bridge change of measure to estimate propagators: paths are generated under an auxiliary diffusion whose bridge between the two fixed boundary points is easy to generate, and the target propagator is recovered through a Radon-Nikodym derivative or likelihood ratio correction derived from Girsanov's theorem. This method overcomes the main limitations of the crude Monte Carlo estimator, which relies on binning in state space, introducing bias and becoming inefficient with rare transitions. We apply the method to a range of benchmark processes and show that the bridge-based estimator matches the accuracy of crude Monte Carlo with orders of magnitude fewer sample paths. Finally, we elaborate on discretization schemes for the Radon-Nikodym derivative and on approximations to the best-in-class change of measure, both of which increase the efficiency of the method.