Bounds for a constrained optimal stopping problem

Abstract

In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem ${\mathbb{E}^{x,y}[\theta(x_\tau,y_\tau)-\int_0^\tau c(y_s)ds]}$ over stopping times τ subject to a constraint ${\mathbb{E}^{x,y}\tau \leq\beta}$, where x(.) is a geometric Brownian motion coupled with an arbitrary diffusion process y(.), θ(., .) and c(.) are given positive, continuous functions and β > 0 is a fixed constant. The present result is derived from a corresponding Lagrangian dual problem, and using a recent result of Makasu (Seq Anal 27:435–440, 2008). Examples are given to illustrate our main result.