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November 2003 On the properties of $r$-excessive mappings for a class of diffusions
Luis H. R. Alvarez
Ann. Appl. Probab. 13(4): 1517-1533 (November 2003). DOI: 10.1214/aoap/1069786509


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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Luis H. R. Alvarez. "On the properties of $r$-excessive mappings for a class of diffusions." Ann. Appl. Probab. 13 (4) 1517 - 1533, November 2003.


Published: November 2003
First available in Project Euclid: 25 November 2003

zbMATH: 1072.60065
MathSciNet: MR2023887
Digital Object Identifier: 10.1214/aoap/1069786509

Primary: 60G40 , 60H30 , 60J60 , 62L15 , 93E20

Keywords: $r$-excessive mappings , $r$-harmonic mappings , convex inequalities , fundamental solutions , Linear diffusions , Optimal stopping

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.13 • No. 4 • November 2003
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