OPTIMAL INVESTMENT POLICY AND DIVIDEND PAYMENT STRATEGY IN AN INSURANCE COMPANY

• Published in: The Annals of applied probability Vol. 20; no. 4; pp. 1253 - 1302
• Main Authors: Azcue, Pablo, Muler, Nora
• Format: Journal Article
• Language: English
• Published: Hayward Institute of Mathematical Statistics 01-08-2010; The Institute of Mathematical Statistics
• Subjects: 49L25; 91B28; 91B30; 91B70; band strategy; Bankruptcy; barrier strategy; Boundary conditions; Continuous functions; Cramér–Lundberg process; Differential equations; dividend payment strategy; Dividends; dynamic programming principle; Hamilton–Jacobi–Bellman equation; insurance; Insurance companies; Investment policy; Investment strategies; optimal investment policy; Optimal strategies; Payments; risk control; Securities markets; Stochastic models; Viscosity; viscosity solution
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/09-aap643

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramér—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.