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)YdB driven by a Brownian motion and study the monotonicity and continuity in y of the value function v(x,y)=sup.Ex,y[e.q.g(X.)], 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.