On Generating Monte Carlo Samples of Continuous Diffusion Bridges

Diffusion processes are widely used in engineering, finance, physics, and other fields. Usually continuous-time diffusion processes can be observed only at discrete time points. For many applications, it is often useful to impute continuous-time bridge samples that follow the diffusion dynamics and connect each pair of the consecutive observations. The sequential Monte Carlo (SMC) method is a useful tool for generating the intermediate paths of the bridge. The paths often are generated forward from the starting observation and forced in some ways to connect with the end observation. In this article we propose a constrained SMC algorithm with an effective resampling scheme guided by backward pilots carrying the information of the end observation. This resampling scheme can be easily combined with any forward SMC sampler. Two synthetic examples are used to demonstrate the effectiveness of the resampling scheme.