On certain integral functionals of squared Bessel processes

摘要

For a squared Bessel process, X, the Laplace transforms of joint laws of (U, integral(Ry)(0) X-s(p) ds) are studied where R-y is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until R-y. A subset of these results are then used to solve the associated small ball problems for integral(Ry)(0) X-s(p) ds ds and to determine a Chung's law of the iterated logarithm. (integral(Ry)(0) X-s(p) ds) is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and to determine the asymptotics for the prices of some put options that are only slightly in-the-money.