🤖 AI Summary
This work addresses the efficient simulation of diffusion bridges—diffusion paths conditioned on fixed endpoints—a critical task in statistical inference for discretely observed diffusion processes. Existing methods suffer from low accuracy and poor numerical stability. To overcome these limitations, we propose the first framework that incorporates score matching into time-reversed learning of diffusion bridges, based on a variational principle. Our approach unifies backward time-reversal dynamics with the forward Doob *h*-transform representation via a bidirectional approximation scheme. Integrating score matching, variational inference, stochastic differential equation (SDE) time reversal, and robust numerical solvers, the method is validated on the Ornstein–Uhlenbeck process, interest-rate financial models, and a genetic model of cellular differentiation. Results demonstrate substantial improvements in bridge-path sampling accuracy and computational stability. The proposed framework provides a scalable, robust tool for Bayesian inference of diffusion processes under discrete observations.
📝 Abstract
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 dis-cuss 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.
