An optimal reinsurance problem in the Cramèr-Lundberg model with investment and transaction costs: An optimal reinsurance problem in the Cramèr-Lundberg model

Abstract

We study optimal proportional reinsurance strategies for minimizing the probability of ruin in an extended Cram\'{e}r--Lundberg risk model with investment returns and fixed transaction costs. Under the assumption of cheap proportional reinsurance, stochastic control methods are used to derive the Hamilton--Jacobi--Bellman equation, reformulated as a Volterra integral equation of the second kind.To obtain tractable solutions, a block-by-block numerical scheme based on Simpson's rule is developed to approximate survival probabilities under light-tailed and heavy-tailed claim distributions.Ruin probabilities increase with retention, transaction costs, and volatility, and decrease with higher investment returns, reflecting strong sensitivity to financial and operational frictions, as confirmed numerically: increasing returns from 10\% to 30\% reduces the ruin probability at $u = 5$ from 0.2319 to 0.1626 under mixed exponential claims, while raising transaction costs from $k = 0$ to $k = 1$ increases it from approximately 0.09 to 0.15 under Pareto$(3,2)$ claims.The optimal retention level is $\alpha^* = 0$, although in practice a small positive retention may be more realistic.