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...

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Bibliographic Details
Published in:Applied mathematics & optimization Vol. 73; no. 2; pp. 271 - 312
Main Authors: Chiarolla, Maria B., De Angelis, Tiziano
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2016
Springer Nature B.V
Subjects:
Approximation
Calculus of Variations and Optimal Control; Optimization
Control
Diffusion
Drift
Hilbert space
Inequalities
Inequality
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Optimization
Partial differential equations
Pricing
Simulation
Stochastic models
Studies
Systems Theory
Theoretical
Viscosity
Optimal stopping
Infinite-dimensional stochastic analysis
60G40
49J40
Degenerate variational inequalities
35R15
Parabolic partial differential equations
ISSN:0095-4616, 1432-0606
Online Access:Get full text
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Summary: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 (Application of variational inequalities in stochastic control, 1982 ), 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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ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-015-9302-8