Some perpetual integral functionals of the three-dimensional Bessel process
Some perpetual integral functionals of the three-dimensional Bessel process
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三维贝塞尔过程的一些永积分泛函
DOI：
10.1142/s0219493723500089
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发表时间：
2023
影响因子：
1.1
通讯作者：
Yukihiro Tsuzuki
中科院分区：
文献类型：
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作者：
Yukihiro Tsuzuki
我们用修正贝塞尔函数、高斯超几何函数和合流超几何函数来计算三维贝塞尔过程的一些积分函数的拉普拉斯变换。得到了一些新的结果，并恢复了一些已确定的结果，如杜弗雷纳的永续性及其翻译版本的一个特殊情况。特别地，我们导出了二维随机变量(∫0∞(exp（ρ t) - η) - 1dt,∫0∞(exp(ρ t) - η) - 2dt）的拉普拉斯变换，其中是三维贝塞尔过程和。推导的一个关键步骤是构造一个关于扩大过滤的鞅，就像在Skorokhod嵌入问题的Bass解中所做的那样。
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 (∫0∞(exp(ρ t) − η)−1dt,∫ 0∞(exp(ρ t) − η)−2dt), whereis a three-dimensional Bessel process and. 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.