Variational inequalities in Hilbert spaces with measures and optimal stopping problems

We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted $L^2$ setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.Comment: To appear in Applied Mathematics and Optimizatio

A Simulation Approach to Optimal Stopping Under Partial Information

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

Optimal stopping in Hilbert spaces and pricing of American options

We consider an optimal stopping problem for a Hilbert-space valued diffusion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial differential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield curve dynamics. Key words. Optimal stopping, obstacle problems, viscosity solutions, option pricing. AMS Subject Classification. 35R20, 49L25, 90A09, 49J15, 60H10. 1 Introduction Optimal stopping problems in finite dimensional domains and obstacle partial differential equations associated with them have been studied extensively in the past. Classical results Andrzej ' Swi¸ech was partially supported by NSF grant DMS-9706760 on the subject can be found in [2],[1]. In recent years the field of stochastic optimal control has developed significantly due to the introduction of the notion of ..