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Optimal stopping of a Hilbert space valued diffusion

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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 \sigma(X) and a generic unbounded operator A in the drift term. When the gain function \Psi is time-dependent and fulfills mild regularity assumptions, the value function V of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient\sigma(X) is specified, the solution of the variational problem is found in a suitable Banach space B fully characterized in terms of a Gaussian measure \mu. This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions [1], 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.

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