Variational inequalities in Hilbert spaces with measures and optimal stopping problems

2006/08/15 by Viorel Barbu, Barbu, Viorel, Carlo Marinelli +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #35K85 #35Q80 #74S05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications #math.AP #math.PR #msc:35K85 #msc:35Q80 #msc:74S05

paper · pdf · doi:10.48550/arxiv.math/0608379

To appear in Applied Mathematics and Optimization

openalex publication_date 2006/08/15 · arxiv created 2007/08/17 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the existence theory for parabolic variational inequalities in weighted L² spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L² setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

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