Optimal Investment Strategy to Maximize the Expected Utility of an Insurance Company Under the Cramer-Lundberg Dynamic

In this work, it is examined the combined problem of optimal portfolio selection rules for an insurer in a continuous-time model where the surplus of an insurance company is modelled as a compound Poisson process. The company can invest its surplus in a risk free asset and in a risky asset, under the Black-Scholes equation. According to the Utility theory, in any financial market where investors face uncertainty, they are not concerned about wealth maximization per se but with utility maximization. It is therefore possible to introduce an increasing and concave utility function φ(x,t) representing the expected utility of a risk-averse investor (for instance, an insurance company). Therefore, the goal of this research work is no longer to maximize the expected portfolio value, minimize the ruin probability or even maximize the expectation of the present value of all dividends paid to the shareholders up to the ruin, but to maximize the expected utility stemming from the wealth during the life insurance contract [0,T]. In this direction, using the Dynamic Programming Principle of the problem, we obtain the Hamilton–Jacobi– Bellman equation by our optimization problem (HJB). Finally, in some cases, we present numerical solutions obtaining the Merton’s strategy as the optimal one