The Solution to a Differential-Difference Equation Arising in Optimal Stopping of a Jump-Diffusion Process

Abstract

In this paper we present a solution to a second order differential–difference equation that occurs in different contexts, specially in control engineering and finance. This equation leads to an ordinary differential equation, whose homogeneous part is a Cauchy–Euler equation. We derive a particular solution to this equation, presenting explicitly all the coefficients. The differential–difference equation is motivated by investment decisions addressed in the context of real options. It appears when the underlying stochastic process follows a jump-diffusion process, where the diffusion is a geometric Brownian motion and the jumps are driven by a Poisson process. The solution that we present - which takes into account the geometry of the problem - can be written backwards, and therefore its analysis is easier to follow.