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

TL;DR

This paper investigates the monotonicity and continuity of the value function in a two-dimensional optimal stopping problem involving stochastic processes, with applications to American option pricing under stochastic volatility models.

Contribution

It establishes monotonicity and continuity properties of the value function for a class of two-dimensional stochastic processes, extending the understanding of optimal stopping in financial models.

Findings

Proves monotonicity of the value function in the variable y.

Shows continuity of the value function in y.

Applies results to American option pricing models with stochastic volatility.

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.