Simulating Diffusion Bridges using the Wiener Chaos Expansion

In this paper, we present a simulation method for diffusion bridges by utilizing an approximation of the Wiener-Chaos Expansion (WCE) applied to a related diffusion process. Indeed, we consider the solution of a proposed diffusion bridge defined through a stochastic differential equation (SDE), and apply the WCE to this solution. As a result, we have developed a rapid method for simulating proposal diffusion bridges that eliminates the need for rejection rates. The method presented in this work could be beneficial in statistical inference, where many parameters are unknown and do not have associated information. This initial uncertainty can influence the effectiveness of data augmentation methods used in inference and increase their rejection rates. In opposition, our method excels by reliably constructing diffusion bridges quickly and efficiently, regardless of initial parameter conditions. We validate our method with a simple Ornstein-Uhlenbeck process. We have also compared our method with two existing methods and determined that its performance is satisfactory. We apply our method to four examples of SDEs and show the numerical results.