Asymptotic and Numerical Analysis of the Optimal Investment Strategy for an Insurer

The asymptotic behaviour of the optimal investment strategy for an insurer is analysed for a number of cash flow processes. The insurer's portfolio consists of a risky stock and a bond and the cash flow is assumed to be either a normal or a compound Poisson process. For a normally distributed cash flow, the asymptotic limits are found in the cases that the stock is very risky or very safe. For a compound Poisson risk process, a composite asymptotic expansion is found for the optimal investment strategy in the case that stock is very risky and the claim size distribution is of an exponential type. In general for a compound Poisson cash flow, the outer asymptotic limit reduces the integro-differential equation describing the optimal stock allocation to an integral equation, which determines the classical survival probability in ruin theory. The leading order optimal asset allocation is derived from this survival probability through a feedback law. Calculation of the optimal asset allocation leads to a difficult numerical problem because of the boundary layer structure of the solution and the tail properties of the claim size distribution. A second order numerical method is successfully developed to calculate the optimal allocation for light and heavy-tailed claim size distributions