Asymptotics of optimal investment behavior under a risk process with two-sided jumps

We study an optimal investment control problem for an insurance company having two business branches, life annuity insurance and non-life insurance. The company can invest its surplus into a risk-free asset and a risky asset with the price dynamics given by a geometric Brownian motion. The optimization objective is to maximize the survival probability of the total portfolio over the infinite time interval. In the absence of investments, the portfolio surplus is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes, and upward jumps are interpreted as random gains that arise at the final moments of the life annuity contracts' realizations (i.e., at the moments of death of policyholders) as a result of the release of unspent funds. The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. The solving to the optimization problem that yields the maximal survival probability, as well as the optimal strategy, is related to the classical solution of the corresponding Hamilton – Jacobi – Bellman (HJB) equation, if this solution exists. In the considered risk model, HJB includes integral operators of two types: Volterra and non-Volterra ones. The presence of the latter makes the asymptotic analysis of the solution quite complicated. However, for the case of small jumps (when the jumps have exponential distributions), we obtained asymptotic representations of solutions for both small and large values of the initial surplus.

insurance, two-sided jumps, optimal investments, risky asset, survival probability, Hamilton – Jacobi – Bellman equation