Universität zu Köln

Brinker, Leonie Violetta (2021). Stochastic Optimisation of Drawdowns via Dynamic Reinsurance Controls. PhD thesis, Universität zu Köln.

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Abstract

In this work, we analyse optimisation problems related to the minimisation of severity and duration of relative losses, so-called drawdowns. We model the accumulated surplus (total income minus expenses) of an insurance company by a stochastic process X. The `drawdown’ of the surplus is defined as the absolute distance to its running maximum and is a time- and performance-adjusted measure of risk. The drawdown process is a stochastic process with reflection. We consider value functions based on the minimisation of the ‘expected time with critical drawdown’, i.e. the average time during which the drawdown exceeds a critical threshold d>0, by dynamic proportional reinsurance controls. The first chapter contains a detailed explanation of the motivation of drawdown minimisation for insurance companies and in stochastic control theory. In the second chapter, we consider the Cramér-Lundberg model. We show that the minimal expected time in critical drawdown is the unique solution to a Hamilton-Jacobi-Bellman equation by considering a set of generalised discounted penalty functions of Gerber-Shiu type. In the third chapter, we prove that the minimal expected time in critical drawdown and optimal strategy for the diffusion model have explicit representations in terms of the Lambert W function. From these two chapters, we conclude that optimal reinsurance minimising drawdowns stabilises the surplus close to its running maximum. Especially for insurance companies, this enhanced predictability is favourable. By analysing optimally controlled processes, we discover, however, that growth of the running maximum is impeded. This can be a drawback from an economic perspective. In the fourth chapter, we therefore introduce a modified value function, including dividends as an ‘incentive to grow’, and solve the resulting problem for a diffusion surplus model. In the fifth and last chapter, we give an outlook on the various possibilities for further research related.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Brinker, Leonie Violetta lvbrinker@gmail.com UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-549634
Date:	11 November 2021
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Optimal stochastic control, dynamic programming, drawdowns, path functionals, stochastic optimisation in continuous time, reflected stochastic differential equations, piecewise deterministic Markov processes, Hamilton-Jacobi-Bellman equations, stochastic analysis, applied probability, ruin theory, insurance, reinsurance English
Date of oral exam:	24 January 2022
Referee:	Name Academic Title Schmidli, Hanspeter Prof. Dr. Dr. Drewitz, Alexander Prof. Dr. Albrecher, Hansjörg Prof. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/54963