Generation of Conditional Densities in Nonlinear Filtering for Infinite-Dimensional Systems

Consider a signal process that takes values in an infinite-dimensional space and assume that this process is observed through a nonlinear diffusion process. In the nonlinear filtering theory literature, recursive expressions for such problems in the form of conditional expectations and, consequently, measure-valued stochastic partial differential equations on the underlying infinite-dimensional space are obtained. In this work, state estimation problems for such stochastic systems are considered, and nonlinear filtering equations in the form of conditional densities are obtained. This is achieved by first deriving the filtering equations in the form of conditional expectations for a Banach space-valued state process whose dynamics depends on a finite-dimensional Brownian motion. Next, a recursion for the conditional density, which we assume to exist with respect to a σ-finite measure, is obtained. For such an analysis, we obtain the adjoint operator for the generator of the state process, which requires to use the theory of integration by parts in the infinite-dimensional domains. Finally, we discuss some potential application of the derived filtering equations in a portfolio optimization problem for portfolios, which consist of bonds whose price curve is interpreted as a stochastic partial differential equation taking values in a Banach space. Applications in mean field games are also demonstrated.