Continuity properties and the support of killed exponential functionals

[摘要] For two independent Levy processes xi and eta and an exponentially distributed random variable tau with parameter q > 0, independent of xi and eta, the killed exponential functional is given by Vq, xi, eta := integral(tau)(0)e(-xi s-) d eta s. Interpreting the case q = 0 as tau = infinity, the random variable Vq, xi, eta is a natural generalisation of the exponential functional integral(infinity)(0) e(-xi)s(-) d eta s, 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, xi,eta arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised OrnsteinUhlenbeck 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 integral(tau)(0)e(-xi s-) d eta s for fixed t >= 0, as well as for integrals of the form integral(infinity)(0) f (s) d eta s for deterministic functions f. Furthermore, applying the same techniques to the case q = 0, new results on the absolute continuity of the improper integral integral(tau)(0)e(-xi s-) d eta s are derived. (C) 2021 ElsevierB.V. All rights reserved.

[发布日期] 2021-10-01

[关键词] Generalised Ornstein-Uhlenbeck process;Exponential functional;Levy processes;Killing;Absolute continuity;Support