  We study the numerical solution of nonlinear partially observed optimal
stopping problems. The system state is taken to be a multi-dimensional
diffusion and drives the drift of the observation process, which is another
multi-dimensional diffusion with correlated noise. Such models where the
controller is not fully aware of her environment are of interest in applied
probability and financial mathematics. We propose a new approximate numerical
algorithm based on the particle filtering and regression Monte Carlo methods.
The algorithm maintains a continuous state-space and yields an integrated
approach to the filtering and control sub-problems. Our approach is entirely
simulation-based and therefore allows for a robust implementation with respect
to model specification. We carry out the error analysis of our scheme and
illustrate with several computational examples. An extension to discretely
observed stochastic volatility models is also considered.
