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.
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