Interpolation for partly hidden diffusion processes

Abstract

Let X t be n-dimensional diffusion process and S t be a smooth set-valued function. Suppose X t is invisible when X t ∈ S t , but we can see the process exactly otherwise. Let X t 0 ∈ S t 0 and we observe the process from the beginning till the signal reappears out of the obstacle after t 0. With this information, we evaluate the estimators for the functionals of X t on a time interval containing t 0 where the signal is hidden. We solve related 3 PDEs in general cases. We give a generalized last exit decomposition for n-dimensional Brownian motion to evaluate its estimators. An alternative Monte Carlo method is also proposed for Brownian motion. We illustrate several examples and compare the solutions between those by the closed form result, finite difference method, and Monte Carlo simulations.