2012/07/03 by Maria B. Chiarolla, M. B. Chiarolla, Chiarolla, M. B. +3
Economics, Econometrics and Finance · Mathematics · #35R15 #49J40 #60G40 #Advanced Harmonic Analysis Research #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications #math.OC #math.PR #msc:35R15 #msc:49J40 #msc:60G40
paper · pdf · doi:10.48550/arxiv.1207.0720
33 pages; improved exposition, added an example
openalex publication_date 2012/07/03 · arxiv created 2015/01/31 · arxiv updated 2015/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeBen-Lio82, 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.