Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches

Philipp Lukas Strietzel and Henriette Elisabeth Heinrich
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Philipp Lukas Strietzel: Institut für Mathematische Stochastik, Technische Universität Dresden, 01062 Dresden, Germany
Henriette Elisabeth Heinrich: Institut für Mathematische Stochastik, Technische Universität Dresden, 01062 Dresden, Germany

Risks, 2022, vol. 10, issue 6, 1-23

Abstract: We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton–Jacobi–Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions, we show that the optimal strategy lies within a certain subclass of all admissible strategies and reduce the two-dimensional control problem to a one-dimensional one. The results are illustrated by a numerical example and Monte Carlo simulated value functions.

Keywords: admissibility; compound Poisson process; degenerate risk model; dividends; dynamic programming principle; Hamilton–Jacobi–Bellman equation; optimal strategy; simultaneous ruin; stochastic control; viscosity solution (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2022
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