Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm

Abstract

This paper deals with the dividend optimization problem for a financial or an insurance entity which can control its business activities, simultaneously reducing the risk and potential profits. It also controls the timing and the amount of dividends paid out to the shareholders. The objective of the corporation is to maximize the expected total discounted dividends paid out until the time of bankruptcy. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes a mixed classical-impulse stochastic control problem. The analytical part of the solution to this problem is reduced to quasivariational inequalities for a second-order nonlinear differential equation. We solve this problem explicitly and construct the value function together with the optimal policy. We also compute the expected time between dividend payments under the optimal policy.

Optimality results for dividend problems in insurance

“…In an insurance context the introduction of transaction costs charging the dividend payments seems to be relatively new (although there is an early discussion by Porteus [101]) and up to now mostly problems in a diffusion setup are solved, see for example Jeanblanc-Picqué & Shiryaev [73], Paulsen [97] and Cadenillas et al [32,33]. For the compound Poisson risk reserve process, the effect of transaction costs on the optimal control problem was recently investigated in Thonhauser & Albrecher [119].…”

“…Such a dividend strategy naturally appears for diffusion risk reserve processes and transaction costs for dividend payments (cf. Jeanblanc-Piqué & Shiryaev [73] for a simple diffusion model with constant drift and volatility, Cadenillas et al [33] for a mean-reverting diffusion process, Paulsen [97] for general diffusion processes; Cadenillas et al [32] also take proportional reinsurance into account). For risk models with jumps and an impulse strategy of the above type, the literature is still scarce.…”

Mixed Band Control of Mutual Proportional Reinsurance

“…The derivation of the value function is similar to [3] and [14].Suppose that   V x satisfies all of the QVI conditions: (2.12), (2.13) and (2.14). First note that the function   V x is a decreasing function of x , and thus 0 V   .…”

“…However, if bankruptcy is allowed and no intervention is initiated when the process reaches 0, then the boundary condition at 0 becomes straightforward:   0 V 0  (see Cadenillas and Zapatero [3] and Cadenillas, et al [14]). In our case, whether 0 is the point that corresponds to the intervention in the form of a contingent call or whether it corresponds to bankruptcy is not given a priori; rather it is part of the solution to the problem.…”

“…This type of model is considered in the framework of the so-called impulse control. Because we also have a control whose application changes the drift and the diffusion coefficient of the controlled process, the resulting mathematical problem becomes a mixed regular-impulse control problem (e.g., Cadenillas et al [14]). In the case of a pure impulse control, the optimal policy is of the   , , , a A B b type, where the four parameters used to construct the optimal control must be computed as a part of a solution to the problem (see Cadenillas and Zapatero [15], Constantinides and Richard [10], Harrison and Taylor [16], and Paulsen [17]).…”