On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

Abstract

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

A general approximation method for optimal stopping and random delay

“…A major difficulty is in identifying the shape of the stopping set D . Most existing results are concentrated in the cases where D is a semi‐infinite interval; in other words, the optimal policy is “threshold‐type” or “one‐sided”, as the decision maker should stop when the underlying process crosses a one‐sided threshold (e.g., Mordecki, 2002; Boyarchenko & Levendorskii, 2002; Sheu & Tsai, 2012; Mishura & Tomashyk, 2013; Mordecki & Mishura, 2016; Christensen & Irle, 2019; Lin & Yao, 2019). Once it is proven that the one‐sided stopping policy is optimal, the problem is reduced to a single variate optimization to find the optimal threshold.…”

One-sided solutions for optimal stopping problems with logconcave reward functions

“…On the other hand, by imposing more structures on {X t }, the problem (1.1) can be solved explicitly for more general reward functions. Indeed, assuming that {X t } is a matrix-exponential jump diffusion, Sheu and Tsai [19] have obtained an explicit one-sided solution of (1.1) for a fairly general class of increasing and logconcave reward functions g ≥ 0 which satisfy some additional technical conditions.…”