Optimal stopping for Levy processes with polynomial rewards

TL;DR

This paper provides explicit solutions for infinite horizon optimal stopping problems involving Levy processes with polynomial rewards, utilizing the supremum of the process and generalizing averaging functions, with practical examples for quadratic, cubic, and quartic cases.

Contribution

It introduces a method to explicitly solve optimal stopping problems for Levy processes with polynomial rewards using a generalized averaging function approach.

Findings

Explicit solutions for polynomial reward cases are derived.

Examples include quadratic, cubic, and quartic polynomials with Brownian and Kou's processes.

The averaging function can be directly computed for polynomial rewards.

Abstract

Explicit solution of an infinite horizon optimal stopping problem for a Levy processes with a polynomial reward function is given, in terms of the overall supremum of the process, when the solution of the problem is one-sided. The results are obtained via the generalization of known results about the averaging function associated with the problem. This averaging function can be directly computed in case of polynomial rewards. To illustrate this result, examples for general quadratic and cubic polynomials are discussed in case the process is Brownian motion, and the optimal stopping problem for a quartic polynomial and a Kou's process is solved.