Monotonicity of the value function for a two-dimensional optimal stopping problem

Abstract

We consider a pair $(X,Y)$ of stochastic processes satisfying the equation $dX=a(X)YdB$ driven by a Brownian motion and study the monotonicity and continuity in $y$ of the value function $v(x,y)={\sup }_{\tau }{E}_{x,y}[e^{-q\tau }g(X_{\tau })]$, where the supremum is taken over stopping times with respect to the filtration generated by $(X,Y)$. Our results can successfully be applied to pricing American options where $X$ is the discounted price of an asset while $Y$ is given by a stochastic volatility model such as those proposed by Heston or Hull and White. The main method of proof is based on time-change and coupling.

Published in at http://dx.doi.org/10.1214/13-AAP956 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)