A diffusion model for optimal dividend distribution for a company with constraints on risk control.

*(English)*Zbl 1084.91047The authors consider a company whose business activities are modeled by a control process \(a_t\), \(t\geq0\), which takes on values in the interval \([\alpha, \beta]\), \(0\leq \alpha<\beta < +\infty\), with risk and potential profit at any time \(t\) proportional to \(a_t\). The restriction \(\alpha > 0\) reflects the fact that there are institutional or statutory reasons (e.g., the company is public) that its business activities cannot be reduced to zero, unless the company faces bankruptcy. In addition, in this model the company has a constant rate of liability payments, such as mortgage payments on its property or amortization of bonds.

In the case of an insurance company, when the control parameter lies within \([0,1]\) this problem was considered by M. J. Taksar and X. Y. Zhou [Insurance Math. Econom. 22, 105-122 (1998; Zbl 0907.90101)]. In this regard, the model treated by Taksar and Zhou can be viewed as a limiting case of \(\alpha\to 0+\) and \(\beta = 1\). However, the strictly positive lower bound treated in this paper renders the argument of Taksar and Zhou invalid and imposes a great difficulty for the problem. It is interesting to observe that the analytic expression for the optimal return function (value function) obtained in this paper, in the limiting case of \(\alpha\to 0+\), \(\beta = 1\), looks completely different from that in the paper of Taksar and Zhou. However, the authors show via a detailed analysis that these are two analytic expressions for one and the same function.

The authors start the analysis with the value function \(v\) of the underlying stochastic control model. It is shown that \(v\) is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which is interesting in its own right as the underlying stochastic control model is of a mixed regular-singular type. Based on this fact, along with the concavity of the value function, it is proved a priori that \(v\) must be twice continuously differentiable (and hence, as a by-product, must be a classical solution to the HJB equation). The proof is very general and should be applicable to a large set of problems whenever concavity can be proved in advance. Afterwards the authors perform a delicate analysis on the HJB equation, which leads to explicit expressions of the value function for all the possible parameter values. Once this is done, optimal risk control and dividend policies are constructed via the verification theorem and the solution to a Skorohod problem.

In the case of an insurance company, when the control parameter lies within \([0,1]\) this problem was considered by M. J. Taksar and X. Y. Zhou [Insurance Math. Econom. 22, 105-122 (1998; Zbl 0907.90101)]. In this regard, the model treated by Taksar and Zhou can be viewed as a limiting case of \(\alpha\to 0+\) and \(\beta = 1\). However, the strictly positive lower bound treated in this paper renders the argument of Taksar and Zhou invalid and imposes a great difficulty for the problem. It is interesting to observe that the analytic expression for the optimal return function (value function) obtained in this paper, in the limiting case of \(\alpha\to 0+\), \(\beta = 1\), looks completely different from that in the paper of Taksar and Zhou. However, the authors show via a detailed analysis that these are two analytic expressions for one and the same function.

The authors start the analysis with the value function \(v\) of the underlying stochastic control model. It is shown that \(v\) is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which is interesting in its own right as the underlying stochastic control model is of a mixed regular-singular type. Based on this fact, along with the concavity of the value function, it is proved a priori that \(v\) must be twice continuously differentiable (and hence, as a by-product, must be a classical solution to the HJB equation). The proof is very general and should be applicable to a large set of problems whenever concavity can be proved in advance. Afterwards the authors perform a delicate analysis on the HJB equation, which leads to explicit expressions of the value function for all the possible parameter values. Once this is done, optimal risk control and dividend policies are constructed via the verification theorem and the solution to a Skorohod problem.

Reviewer: Alexandr B. Vasil’ev (Odessa)