• Xin Xu

Student thesis: Phd


We consider two optimal stopping problems driven by Levy processes with Omega killing, which can be understood as the so-called bankruptcy event for an insurance company. The main question is to decide when to stop the observation before bankruptcy happens in order to obtain the best reward. We work with stable processes and Cramer-Lundberg processes respectively under different choices of the gain (payoff) functions and time-dependent killing processes. The first problem is the optimal stopping for an Omega-killed stable process. The gain function is similar to an American call option. We show that under certain conditions, the solution to the problem driven by a stable process can be transformed from the one driven by a special type of Levy process. To see this, we first construct, from the original stable process, a positive self-similar Markov process (pssMp) and then, we generate, from this pssMp, a Levy process via Lamperti transform, which is verified to be of the so-called double hypergeometric class. After solving the related optimal stopping problem under such double hypergeometric Levy process, we show that these results are also valid and can be interpreted for the original problem with the stable process. We also share a remark on a similar optimal stopping problem while the gain function is of American put option style and a Markov additive process is considered with the help of Lamperti-Kiu transform. In the second project, we study the optimal stopping problem driven by a Cramer-Lundberg process with piecewise constant killing intensity. The payoff function has a constant penalty p for negative values and is not continuous at zero, which makes it harder to apply the change of measure formula and to follow the classic verification steps of solving optimal stopping problems. Under some mild conditions w.r.t. penalty p in terms of the parameters for Cramer-Lundberg processes, the solutions are fully characterised where the optimal up-crossing thresholds are explicitly defined. The proofs consist of massive calculations based on existing explicit expressions for fluctuation functions of Cramer-Lundberg processes. By introducing a number of lemmas, we solve the optimal stopping problem. We also give numerical examples to illustrate our result and make discussions for future direction.
Date of Award31 Dec 2022
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorKees Van Schaik (Supervisor) & Xiong Jin (Supervisor)


  • Levy processes
  • Optimal stopping
  • Omega killing

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