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Continuity properties and the support of killed exponential functionals
Stochastic Processes and their Applications ( IF 1.4 ) Pub Date : 2021-06-11 , DOI: 10.1016/j.spa.2021.06.002
Anita Behme , Alexander Lindner , Jana Reker , Victor Rivero

For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter q>0, independent of ξ and η, the killed exponential functional is given by Vq,ξ,η0τeξsdηs. Interpreting the case q=0 as τ=, the random variable Vq,ξ,η is a natural generalisation of the exponential functional 0eξsdηs, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional Vq,ξ,η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of 0teξsdηs for fixed t0, as well as for integrals of the form 0f(s)dηs for deterministic functions f. Furthermore, applying the same techniques to the case q=0, new results on the absolute continuity of the improper integral 0eξsdηs are derived.



中文翻译:

连续性和对终止指数泛函的支持

对于两个独立的 Lévy 过程 ξη 和一个指数分布的随机变量 τ 带参数 q>0, 独立于 ξη,被杀死的指数函数由下式给出 q,ξ,η0τ电子-ξ-dη. 案例解读q=0 作为 τ=, 随机变量 q,ξ,η 是指数函数的自然推广 0电子-ξ-dη,其定律在文献中得到了充分研究,因为它是广义 Ornstein-Uhlenbeck 过程的平稳分布。在本文中,我们还证明了杀死指数函数的定律q,ξ,η作为随机微分方程解的平稳分布出现,因此与广义 Ornstein-Uhlenbeck 过程建立了密切联系。此外,表征了杀指数泛函定律的支持性和连续性,推导出了许多绝对连续性的充分条件。我们还得到了绝对连续性的各种新的充分条件0电子-ξ-dη 对于固定 0,以及对于形式的积分 0F()dη 对于确定性函数 F. 此外,将相同的技术应用于案例 q=0, 关于不当积分的绝对连续性的新结果 0电子-ξ-dη 是派生的。

更新日期:2021-06-23
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