On the properties of $r$-excessive mappings for a class of diffusions

Luis H. R. Alvarez

70 citations0/0 files storedThe Annals of Applied Probability

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Probability and Risk Models

Abstract

We consider the convexity and comparative static properties of a class of $r$-harmonic mappings for a given linear, time-homogeneous and regular diffusion process. We present a set of weak conditions under which the minimal $r$-excessive mappings for the considered diffusion are convex and under which an arbitrary nontrivial $r$-excessive mapping is convex on the regions where it is $r$-harmonic. Consequently, we are able to present a set of usually satisfied conditions under which increased volatility increases the value of $r$-harmonic mappings. We apply our results to a class of optimal stopping problems arising frequently in studies considering the pricing of perpetual American contingent claims and state a set of conditions under which the value function is convex on the continuation region and, consequently, under which increased volatility unambiguously increases the value function and expands the continuation region, thus postponing the rational exercise of the claim.

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References

1994Investment Under Uncertainty.1994Investment under Uncertainty 1987Stochastic Differential Equations: An Introduction with Applications.2005Controlled Markov Processes and Viscosity Solutions 1996Handbook of Brownian Motion — Facts and Formulae 1985Optimal Stopping of One-Dimensional Diffusions 2001Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

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