2004/03/03 by Paavo Salminen, Marc Yor, Salminen, Paavo +1
Mathematics · #60J60 #60J65 #60J70 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60J60 #msc:60J65 #msc:60J70
paper · pdf · doi:10.48550/arxiv.math/0403069
arxiv created 2004/03/03 · arxiv updated 2009/12/01
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 IX_∞(f):=∫0^∞ f(Xt) 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 μ>0, denoted \B(μ)t: t≥ 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 ∫0^∞ f(B(μ)t) \bf 1_\B(μ)t<0\ dt \rm and ∫0^∞ f(B(μ)t) \bf 1_\B(μ)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 rôle 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 %associated to the function ∫0^∞ (a+exp(B(μ)t))-2 dt, which is seen, in the case μ=1/2, to be identical in law with the first hitting time for a Brownian motion with drift μ=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.