An optimal reinsurance management and dividend payout strategy when the insurer's reserve is an Ito-Levy process

We solve the problem of an insurer who decides to optimally allocate a proportion (1-a(t)) of premiums to a re-insurance company (thereby retaining a proportion a(t) of premiums) and who also has to optimally pay dividends c(t) at any time t to shareholders. If the insurer's reserve x(t) is a given Ito-Levy process and the shareholders' preferences are given by a general constant relative risk aversion utility function, we set the problem as a stochastic control problem and solve the resulting HJB equation. Results are discussed in detail and show that it is optimal that premiums and dividends be directly proportional to the reserve. High premium lead to high reserves which in turn leads to high premium payouts. This paper contributes to the rich area of stochastic control and also it helps insurers whose reserves can be modelled, to make technical decisions of how to charge premiums, reinsure liabilities and pay dividends to shareholders.