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

Preprint, Article, Other literature type English OPEN
Assing, Sigurd ; Jacka, Saul D. ; Ocejo, Adriana (2014)
  • Publisher: Institute of Mathematical Statistics
  • Journal: (issn: 1050-5164)
  • Related identifiers: doi: 10.1214/13-AAP956
  • Subject: Optimal stopping | QA | coupling | American option | Mathematics - Optimization and Control | 91G20 | Mathematics - Probability | time-change | 60G40 | stochastic volatility model

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.
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