Optimal stopping problems over a finite or an infinite time horizon for Itô’s diffusion processes described by stochastic differential equations (SDEs) arise in many areas of science, engineering, and finance (see, e.g., Fleming and Soner [FS93], Øksendal [Øks00], Shiryaev [Shi78], Karazas and Shreve [KS91], and references contained therein). The objective of the problem is to find a stopping time τ with respect to the filtration generated by the solution process of the SDE that maximizes or minimizes a certain expected reward or cost functional. The value function of these problems are normally expressed as a viscosity or a generalized solution of Hamilton-Jacobi-Bellman (HJB) variational inequality that involves a second-order parabolic or elliptic partial differential equation in a finite-dimensional Euclidean space.
KeywordsVariational Inequality Function Versus Viscosity Solution Comparison Principle Polynomial Growth
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