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.
Publié le : 2003-11-14
Classification:
Linear diffusions,
fundamental solutions,
$r$-harmonic mappings,
$r$-excessive mappings,
convex inequalities,
optimal stopping,
60J60,
60G40,
62L15,
60H30,
93E20
@article{1069786509,
author = {Alvarez, Luis H. R.},
title = {On the properties of $r$-excessive mappings for a class of diffusions},
journal = {Ann. Appl. Probab.},
volume = {13},
number = {1},
year = {2003},
pages = { 1517-1533},
language = {en},
url = {http://dml.mathdoc.fr/item/1069786509}
}
Alvarez, Luis H. R. On the properties of $r$-excessive mappings for a class of diffusions. Ann. Appl. Probab., Tome 13 (2003) no. 1, pp. 1517-1533. http://gdmltest.u-ga.fr/item/1069786509/