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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 11 open journal systems 


Perpetual Integral Functionals as Hitting and Occupation Times

Paavo Salminen, Abo Akademi, Finland
Marc Yor, Université Pierre et Marie Curie


Abstract
Abstract. Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable function. We are interested in finding conditions on $X$ and $f$ which imply that the perpetual integral functional $$ I^X_infty(f):=int_0^infty f(X_t), dt $$ is identical in law with the first hitting time of a point for some other diffusion. This phenomenon may often be explained using random time change. Because of some potential applications in mathematical finance, we are considering mainly the case when $X$ is a Brownian motion with drift $mu>0,$ denoted ${B^{(mu)}_t: tgeq 0},$ but it is obvious that the method presented is more general. We also review the known examples and give new ones. In particular, results concerning one-sided functionals $$ int_0^infty f(B^{(mu)}_t),{bf 1}_{{B^{(mu)}_t<0}} dtquad {rm and}quad int_0^infty f(B^{(mu)}_t),{bf 1}_{{B^{(mu)}_t>0}} dt $$ are presented. This approach generalizes the proof, based on the random time change techniques, of the fact that the Dufresne functional (this corresponds to $f(x)=exp(-2x)),$ playing quite an important role in the study of geometric Brownian motion, is identical in law with the first hitting time for a Bessel process. Another functional arising naturally in this context is $$ int_0^infty big(a+exp(B^{(mu)}_t)big)^{-2}, dt, $$ which is seen, in the case $mu=1/2,$ to be identical in law with the first hitting time for a Brownian motion with drift $mu=a/2.$ The paper is concluded by discussing how the Feynman-Kac formula can be used to find the distribution of a perpetual integral functional.


Full text: PDF

Pages: 371-419

Published on: June 6, 2005
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Electronic Journal of Probability. ISSN: 1083-6489