Optimal control of risk exposure, reinsurance and investments for insurance portfolios

We consider an insurance company whose insurance business follows a diffusion perturbated classical risk process. Assets can be invested in a risk-free asset or in a risky one, the return of the latter follows an independent diffusion perturbated classical risk process (in finance it is called a jump-diffusion process). The company can dynamically control the proportion of the insurance business that is ceded to reinsurers, and in addition it can dynamically cover parts of its remaining business using excess of loss reinsurance. It can also dynamically control the proportion of the assets it invests in the risky asset. We seek to find the controls that maximize expected utility of assets at a terminal time. We do so for three different utility functions. Let T be a fixed time and Ty be the time of ruin, i.e. the first time assets become nonpositive. Then the utility functions and terminal times are: 1. ) = ( max { 0 , y } ) c ,   0 < c < 1 and terminal time is T∧Ty. ) = ln     y when y>0 and U(y)=−∞ when y≤0. Terminal time is T. e−cy, c>0, and terminal time is T. ns out that in spite of the complexity of the problem, provided there are no complicating constraints on the controls, the problems have rather simple solutions. Our vehicle is the Hamilton–Jacobi–Bellmann equations, and we seek to verify that the suggested solutions are in fact optimal among a fairly large easily verifiable admissible class of controls. We also discuss quantitative and qualitative aspects of the solutions.