THE SOLUTION TO A DIFFERENTIAL-DIFFERENCE EQUATION ARISING IN OPTIMAL STOPPING OF A JUMP-DIFFUSION PROCESS

• Published in: Revstat Vol. 20; no. 1; p. 85
• Main Authors: Nunes, Claudia, Pimentel, Rita
• Format: Journal Article
• Language: English
• Published: Instituto Nacional de Estatistica 01.01.2022; Instituto Nacional de Estatística | Statistics Portugal
• Subjects: Commodity futures; Control engineering; differential; differential equation; Differential equations; Differential–difference equation; diffusion process; jump; jump–diffusion process; Stochastic processes; difference equation; differential equation
• ISSN: 1645-6726; 2183-0371
• DOI: 10.57805/revstat.v20i1.364

* 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. Keywords: * differential-difference equation; differential equation, jump-diffusion process. AMS Subject Classification: * 37H10, 60G40.