On certain integral functionals of squared Bessel processes

For a squared Bessel process, X, the Laplace transforms of joint laws of (U;R Ry0 Xps ds) are studied where Ry is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until Ry. A subset of these results are then used to solve the associated small ball problems for R Ry0 Xpsds and determine a Chung's law of iterated logarithm. (Ry0 Xps 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 determine the asymptotics for the prices of some put options that are only slightly in-the-money.