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 Θ 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 [3], of well-known results on optimal stopping theory and variational inequalities in R n. These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.