A Backward Doubly Stochastic Differential Equation Approach for Nonlinear Filtering Problems

A backward doubly stochastic differential equation (BDSDE) based nonlinear filtering method is considered. The solution of the BDSDE is the unnormalized density function of the conditional expectation of the state variable with respect to the observation filtration, which solves the nonlinear filtering problem through the Kallianpur formula. A first order finite difference algorithm is constructed to solve the BSDES, which results in an accurate numerical method for nonlinear filtering problems. Numerical experiments demonstrate that the BDSDE filter has the potential to significantly outperform some of the well known nonlinear filtering methods such as particle filter and Zakai filter in both numerical accuracy and computational complexity.