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

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

Regime-Switching Jump Diffusion Processes with Countable Regimes: Feller, Strong Feller, Irreducibility and Exponential Ergodicity

This work is devoted to the study of regime-switching jump diffusion processes in which the switching component has countably infinite regimes. Such processes can be used to model complex hybrid systems in which both structural changes, small fluctuations as well as big spikes coexist and are intertwined. Weak sufficient conditions for Feller and strong Feller properties and irreducibility for such processes are derived; which further lead to Foster-Lyapunov drift conditions for exponential ergodicity. Our results can be applied to stochastic differential equations with non-Lipschitz coefficients. Finally, an application to feedback control problems is presented