Stochastic Control of Drawdowns via Reinsurance under Random Inspection
Kira Dudziak, Hanspeter Schmidli
TL;DR
Addresses stabilising an insurer’s surplus by controlling drawdown in a diffusion model with proportional reinsurance, where observations (and retention updates) occur at renewal times; the problem is formulated as minimizing the discounted count of observations in the critical drawdown interval, with DP equation $v(z)=\mathbf{1}_{\{z>d\}}+\inf_{b\in[0,1]} \mathbb{E}[e^{-rT_1} v(\Delta_z^{b}(T_1))]$ and an explicit distribution for $\Delta_z^b(t)$; the paper proves the existence of an optimal policy $B^{*}$ and provides a numerically tractable scheme, including closed-form results for the Poisson and deterministic observation cases; numerics illustrate how the optimal retention $b(z)$ depends on the reinsurance price and on the observation regime, and show convergence to the continuous-time Brinker–Schmidli benchmark as observation frequency increases.
Abstract
We consider a diffusion risk model where proportional reinsurance can be bought. In order to stabilise the surplus process, one tries to keep the drawdown, that is the difference of the surplus to its historical maximum, in an interval $[0,d)$. The observation times of the drawdowns form a renewal process. The retention levels can only be changed at the observation times either. We show that an optimal strategy exists and how it is determined. We illustrate the findings in the case of Poissonian observation times and deterministic inter-observation times.