Bounds for a constrained optimal stopping problem

In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem E-x,E-y [theta (x(tau), y(tau)) - integral(tau)(0) c(y(s))ds] over stopping times tau subject to a constraint E-x,E-y tau <= beta, where x(.) is a geometric Brownian motion coupled with an arbitrary diffusion process y(.), theta(., .) and c(.) are given positive, continuous functions and beta > 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.