Optimal Stopping of a Hilbert Space valued Diffusion: an Infinite Dimensional Variational Inequality

A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space H with a non-linear diffusion coefficient σ(X) and a generic unbounded operator A in the drift term. When the gain function is time-dependent and fulfils mild regularity assumptions, the value function U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient σ(X) is specified, the solution of the variational problem is found in a suitable Banach space V fully characterized in terms of a Gaussian measure μ. This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982), of well-known results on optimal stopping theory and variational inequalities in Rn. These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.