An optimal investment strategy for maximizing the expected value of utility accumulation across capital levels

This paper examines an optimal investment problem for an insurance company under the Cramer–Lundberg risk model, where investments are allocated between risky and risk-free assets. In contrast to models focusing on optimal investment and/or reinsurance strategies to maximize the expected utility of terminal wealth within a given time horizon, this study considers the expected value of utility accumulation across all intermediate capital levels of the insurer. We employ the dynamic programming principle and prove a verification theorem showing that any solution to the Hamilton–Jacobi–Bellman (HJB) equation solves our optimization problem. We establish the existence of the optimal investment strategy subject to some regular conditions for the solution of the HJB equation. Finally, we present numerical examples to illustrate the applicability of the theoretical findings.