# Optimal Surplus-dependent Reinsurance under Regime-Switching in a Brownian Risk Model

*Julia Eisenberg*,
*Lukas Fabrykowski* and
*Maren Diane Schmeck*

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Julia Eisenberg: Center for Mathematical Economics, Bielefeld University

Lukas Fabrykowski: Center for Mathematical Economics, Bielefeld University

Maren Diane Schmeck: Center for Mathematical Economics, Bielefeld University

No 648, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

**Abstract:**
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company's surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching complicates substantially the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton-Jacobi-Bellman equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first and second order derivatives. Via Ito's formula we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation.

**Keywords:** Reinsurance; Regime-switching; Brownian motion; Markov chain; Optimal control; HJB equation; Ordinary differential equations; Boundary value problem (search for similar items in EconPapers)

**Pages:** 36

**Date:** 2021-04-06

**New Economics Papers:** this item is included in nep-rmg

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https://pub.uni-bielefeld.de/download/2953603/2953606 First Version, 2021 (application/pdf)

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**Persistent link:** https://EconPapers.repec.org/RePEc:bie:wpaper:648

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