On certain integral functionals of squared Bessel processes

Abstract

Let $X$ be a squared Bessel process. Following a Feynman-Kac approach, the Laplace transforms of joint laws of $(U,{\int }_0^{R_y}{X}_s^{p}ds)$ 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 ${\int }_0^{R_y}{X}_s^{p}ds$ and determine a Chung's law of iterated logarithm. $({\int }_0^{R_y}{X}_s^{p}ds)$ 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 determine the asymptotics for the prices of some put options that are only slightly in-the-money.

References in corpus (2)

• Analysis of continuous strict local martingales via h-transforms
• Small deviations of general Lévy processes

Cited by in corpus (1)

• Bilateral Credit Valuation Adjustment for Large Credit Derivatives Portfolios