Viscosity Solution for Optimal Stopping Problems of Feller Processes

Abstract

We study an optimal stopping problem when the state process is governed by a general Feller process. In particular, we examine viscosity properties of the associated value function with no a priori assumption on the stochastic differential equation satisfied by the state process. Our approach relies on properties of the Feller semigroup. We present conditions on the state process under which the value function is the unique viscosity solution to an Hamilton-Jacobi-Bellman (HJB) equation associated with a particular operator. More specifically, assuming that the state process is a Feller process, we prove uniqueness of the viscosity solution which was conjectured in [26]. We then apply our results to study viscosity property of optimal stopping problems for some particular Feller processes, namely diffusion processes with piecewise coefficients and semi-Markov processes. Finally, we obtain explicit value functions for optimal stopping of straddle options, when the state process is a reflected Brownian motion, Brownian motion with jump at boundary and regime switching Feller diffusion, respectively (see Section 8).

Risk-sensitive optimal stopping with unbounded terminal cost function

“…Also, it may result from the use of viscosity techniques applied to variational inequalities; see e.g. Bensoussan and Lions (1984) and Dai and Menoukeu-Pamen (2018).…”

Exact optimal stopping for multidimensional linear switching diffusions

“…For multidimensional optimal stopping problems, it is nearly impossible to find a closed form solution for the free-boundary partial differential equation (PDE) in most cases. In these cases, one may resort to excessive functions, Green kernel representations, and/or solutions to Hamilton-Jacobi-Bellman (HJB) equations (see [3,5,6,7,17,18]). A key drawback of these characterizations of the value function is that they do not provide an exact optimal stopping policy.…”

A new approach to optimal stopping for Hunt processes

“…Then one must use the viscosity solution approach to verify that a given function is indeed the value function. This is done in for example Dai & Menoukeu Pamen [2], where the authors use a viscosity solution approach to study optimal stopping of some processes with reflection. More precisely, they prove that the value function is the unique viscosity solution of the HJB equation associated with the optimal stopping problem of reflected Feller processes.…”

“…Box 1053 Blindern, N-0316 Oslo, Norway. Emails: achrefb@math.uio.no, olfad@math.uio.no, oksendal@math.uio.no2 This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.…”