On reinsurance and investment for large insurance portfolios

We consider a problem of optimal reinsurance and investment for an insurance company whose surplus is governed by a linear diffusion. The company’s risk (and simultaneously its potential profit) is reduced through reinsurance, while in addition the company invests its surplus in a financial market. Our main goal is to find an optimal reinsurance–investment policy which minimizes the probability of ruin. More specifically, in this paper we consider the case of proportional reinsurance, and investment in a Black–Scholes market with one risk-free asset (bond, or bank account) and one risky asset (stock). We apply stochastic control theory to solve this problem. It transpires that the qualitative nature of the solution depends significantly on the interplay between the exogenous parameters and the constraints that we impose on the investment, such as the presence or absence of shortselling and/or borrowing. In each case we solve the corresponding Hamilton–Jacobi–Bellman equation and find a closed-form expression for the minimal ruin probability as well as the optimal reinsurance–investment policy.