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## 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 $dX=a(X)Y\,dB$ 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.Comment: 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

Topics: Mathematics - Probability, Mathematics - Optimization and Control
Year: 2014
DOI identifier: 10.1214/13-AAP956
OAI identifier: oai:arXiv.org:1208.3126

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