On certain integral functionals of squared Bessel processes

For a squared Bessel process, , the Laplace transforms of joint laws of are studied where is the first hitting time of by and is a random variable measurable with respect to the history of X until . A subset of these results are then used to solve the associated small ball problems for and to determine a Chung's law of the iterated logarithm. 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.