# Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

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 $$\mathcal {H}$$ H with a non-linear diffusion coefficient $$\sigma (X)$$ σ ( X ) and a generic unbounded operator A in the drift term. When the gain function $$\Theta$$ Θ is time-dependent and fulfils mild regularity assumptions, the value function $$\mathcal {U}$$ U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $$\sigma (X)$$ σ ( X ) is specified, the solution of the variational problem is found in a suitable Banach space $$\mathcal {V}$$ V fully characterized in terms of a Gaussian measure $$\mu$$ 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 $$\mathbb {R}^n$$ R n . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

, Volume 73 (2) – Apr 1, 2016
42 pages

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/lp/springer_journal/optimal-stopping-of-a-hilbert-space-valued-diffusion-an-infinite-BLXhWGGmOL
Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9302-8
Publisher site
See Article on Publisher Site

### Abstract

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 $$\mathcal {H}$$ H with a non-linear diffusion coefficient $$\sigma (X)$$ σ ( X ) and a generic unbounded operator A in the drift term. When the gain function $$\Theta$$ Θ is time-dependent and fulfils mild regularity assumptions, the value function $$\mathcal {U}$$ U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $$\sigma (X)$$ σ ( X ) is specified, the solution of the variational problem is found in a suitable Banach space $$\mathcal {V}$$ V fully characterized in terms of a Gaussian measure $$\mu$$ 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 $$\mathbb {R}^n$$ R n . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2016

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