### Stochastic Optimisation of Drawdowns via Dynamic Reinsurance Controls

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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 = (Xt)t≥0, which is either a Cram´er–Lundberg process or a diffusion. The running maximum M = (Mt)t≥0 and the drawdown Δ = (Δt)t≥0 of X are given by Mt = max{m0, sups∈[0,t] Xs} and Δt = Mt −Xt at time t ≥ 0. With this definition, we allow for an initial maximum m0 ∈ R which has been reached before the observation starts. The drawdown process has the natural interpretation of the current decline from the last historical peak of the surplus and is, therefore, a time- and performance-adjusted measure of risk. We consider value functions based on the minimisation of the ‘expected time with critical drawdown’ ER ∞0 e−δt1{Δt>d} dt Δ0 = x, x ≥ 0, by dynamic, proportional reinsurance controls. Here, the parameter d > 0 is a proxy for the size of drawdowns that is perceived as unfavourable. The ‘discounting’ rate δ > 0 reflects the preference of postponing critical drawdowns for as long as possible.

The first chapter contains a detailed explanation of the motivation of drawdown minimisation for insurance companies and in stochastic control theory. We prove that the problem can be split into the subproblems of

i) maximising the time with uncritical drawdown, Δ ∈ [0, d], with a penalty for the overshoot at the exit time and

ii) minimising the time of recovery if the drawdown is currently critical, Δ > d.

In the second chapter, we consider the Cram´er–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 our numerical examples, we consider in detail the optimal strategies (which are of feedback form in all cases). By putting the focus on a different aspect in each chapter, we highlight model-specific results: in the second chapter, we address the influence of the claim distribution, in the third chapter, the effect of costs of reinsurance and in the fourth chapter, the impact of preference (paying dividends versus avoiding drawdowns) of the insurer. In the fifth and last chapter, we give an outlook on the various possibilities for further research related.

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Categories: ASTIN / NON-LIFE
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