Some Exponential Integral Functionals of BM(μ) and BES(3)

Abstract

In the present paper, we derive the Laplace transforms of the integral functionals

$$\int_0^\infty {\left( {p\left( {\exp \left( {B_t^{\left( \mu \right)} } \right) + 1} \right)^{ - 1} + q\left( {\exp \left( {B_t^{\left( \mu \right)} } \right) + 1} \right)^{ - 2} } \right)} \;dt$$

and

$$\int_0^\infty {\left( {p\left( {\exp \left( {R_t^{\left( 3 \right)} } \right) - 1} \right)^{ - 1} + q\left( {\exp \left( {R_t^{\left( 3 \right)} } \right) - 1} \right)^{ - 2} } \right)} \;dt,$$

where p and q are real numbers, {B ^(μ)_t : t ≥ 0} is a Brownian motion with drift μ > 0 (denoted BM(μ)), and {R ⁽³⁾_t : t ≥ 0} is a 3-dimensional Bessel process (denoted BES(3)). The transforms are given in terms of Gauss' hypergeometric functions, and the results are closely related to some results for functionals of Jacobi diffusions. This work generalizes and completes some results of Donati-Martin and Yor and Salminen and Yor. Bibliography: 18 titles.