We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss’ hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable $({\int }_0^{\infty }{(exp({\rho }_t)-\eta )}^{-1}dt,{\int }_0^{\infty }{(exp({\rho }_t)-\eta )}^{-2}dt),$ where $\rho$ is a three-dimensional Bessel process and $\eta \le 1$. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration.

我们根据修正贝塞尔函数、高斯超几何函数和合流超几何函数计算三维贝塞尔过程的一些积分泛函的拉普拉斯变换。获得了一些新的结果，并恢复了几个既定的结果，例如 Dufresne 的 perpetuity 及其翻译版本的一个特例。特别地，我们推导出二维随机变量的拉普拉斯变换$({\int }_0^{\infty }{(exp({\rho }_{吨})-\eta )}^{-\mathrm{1个}}d吨,{\int }_0^{\infty }{(exp({\rho }_{吨})-\eta )}^{-\mathrm{2个}}d吨),$在哪里$\rho$是三维贝塞尔过程并且$\eta \le \mathrm{1个}$. 推导的一个关键步骤是构建一个鞅，就像在 Skorokhod 嵌入问题的 Bass 解决方案中针对扩大过滤所执行的那样。