Distributional properties of killed exponential functionals and short-time behavior of Lévy-driven stochastic differential equations [thesis]

Jana Reker, Universität Ulm
2021
This thesis covers different properties of solutions to Lévy-driven stochastic differential equations. In Chapter 2, we consider various distributional properties of killed exponential functionals of Lévy processes. Similar to the case without killing, there are two ways to view the distribution of this random variable, leading to two main approaches to studying its properties. First, we consider the killed exponential functional as a stopped stochastic integral. Here, using tools from
more » ... tools from probability theory and infinitely divisible distributions, the support and continuity of the law of the killed exponential functional are characterized, and many sufficient conditions for absolute continuity are given. Additionally, sufficient conditions for absolute continuity of different conditioned integrals and in the case without killing are obtained. Further, it can be shown that the law of the killed exponential functional arises as the stationary distribution of a solution to a stochastic differential equation. Since the solution is closely related to the generalized Ornstein-Uhlenbeck process and, in particular, a Markov process, tools from functional analysis become applicable. In the second part of Chapter 2, the infinitesimal generator of the process is calculated and used to derive different distributional equations describing the law of the killed exponential functional, as well as functional equations for its Lebesgue density in the absolutely continuous case. We then consider different special cases and examples to obtain more explicit information on the law of the killed exponential functional and to illustrate some applications of the equations, which cover the case without killing as well. In Chapter 3, we consider solutions to more general Lévy-driven stochastic differential equations. While the almost sure short-time behavior of Lévy processes is well-known and can be characterized in terms of the generating triplet, there is no complete characterization of the behavior of the solution X. Using methods from st [...]
doi:10.18725/oparu-35350 fatcat:mow2qkueybc5rgwmdirzdemzue