Optimal investment policy and dividend payment strategy in an insurance company

TL;DR

This paper investigates the optimal combination of investment and dividend strategies for an insurance company with a stochastic reserve process, aiming to maximize discounted dividends until bankruptcy, and characterizes the optimal policy and value function.

Contribution

It introduces a comprehensive method to determine optimal investment and dividend policies, including a verification approach and analysis of the value function's regularity and structure.

Findings

Optimal dividend strategy has a band structure.

The value function is the smallest viscosity solution of the HJB equation.

Optimal strategies can be non-barrier and the value function may lack smoothness.

Abstract

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cram\'{e}r--Lundberg process. The firm has the option of investing part of the surplus in a Black--Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton--Jacobi--Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.