Simulating Diffusion Bridges with Score Matching

We consider the problem of simulating diffusion bridges, which are diffusion processes that are conditioned to initialize and terminate at two given states. The simulation of diffusion bridges has applications in diverse scientific fields and plays a crucial role in the statistical inference of discretely-observed diffusions. This is known to be a challenging problem that has received much attention in the last two decades. This article contributes to this rich body of literature by presenting a new avenue to obtain diffusion bridge approximations. Our approach is based on a backward time representation of a diffusion bridge, which may be simulated if one can time-reverse the unconditioned diffusion. We introduce a variational formulation to learn this time-reversal with function approximation and rely on a score matching method to circumvent intractability. Another iteration of our proposed methodology approximates the Doob's $h$-transform defining the forward time representation of a diffusion bridge. We discuss algorithmic considerations and extensions, and present numerical results on an Ornstein--Uhlenbeck process, a model from financial econometrics for interest rates, and a model from genetics for cell differentiation and development to illustrate the effectiveness of our approach.