A worst-case optimization approach to impulse perturbed stochastic control with application to financial risk management

This work presents the main ideas, methods and results of the theory of impulse perturbed stochastic control as an extension of the classic stochastic control theory. Apart from the introduction and the motivation of the basic concept, two stochastic optimization problems are the focus of the investigations. On the one hand we consider a differential game as analogue of the expected utility maximization problem in the situation with impulse perturbation, and on the other hand we study an appropriate version of a target problem. By dynamic optimization principles we characterize the associated value functions by systems of partial differential equations (PDEs). More precisely, we deal with variational inequalities whose single inequalities comprise constrained optimization problems, where the corresponding admissibility sets again are given by the seeked value functions. Using the concept of viscosity solutions as weak solutions of PDEs, we avoid strong regularity assumptions on the value functions. To use this concept as sufficient verification method, we additionally have to prove the uniqueness of the solutions of the PDEs. As a second major part of this work we apply the presented theory of impulse perturbed stochastic control in the field of financial risk management where extreme events have to be taken into account in order to control risks in a reasonable way. Such extreme scenarios are modelled by impulse controls and the financial decisions are made with respect to the worstcase scenario. In a first example we discuss portfolio problems as well as pricing problems on a capital market with crash risk. In particular, we consider the possibility of trading options and study their in uence on the investor's performance measured by the expected utility of terminal wealth. This brings up the question of crash-adjusted option prices and leads to the introduction of crash insurance. The second application concerns an insurance company which faces potentially large losses from extreme damages. We propose a dynamic model where the insurance company controls its risk process by reinsurance in form of proportional reinsurance and catastrophe reinsurance. Optimal reinsurance strategies are obtained by maximizing expected utility of the terminal surplus value and by minimizing the required capital reserves associated to the risk process