Vol. 41, No. 6, pp. 1946-1979

Abstract. This paper investigates a model of a corporation which faces constant liability payments and which can choose a production/business policy from an available set of control policies with different expected profits and risks. The objective is to find a business policy and a dividend distribution scheme so as to maximize the expected present value of the total dividend distributions. The main feature of this paper is that there are constraints on business activities such as inability to completely eliminate risk (even at the expense of reducing the potential profit to zero) or when such a risk cannot exceed a certain level. The case in which there is no restriction on the dividend pay-out rates is dealt with. This gives rise to a mixed regular-singular stochastic control problem. First the value function is analyzed in great detail and in particular is shown to be a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Based on this it is further proved that the value function must be twice continuously differentiable. Then a delicate analysis is carried out on the HJB equation, leading to an explicit expression of the value function as well as the optimal policies.

Key words, diffusion model, dividend distribution, risk control, optimal stochastic control, HJB equation, viscosity solution, Skorohod problem

1. Introduction. Recently there has been an upsurge of interest in diffusion models for optimal dividend optimization and/or risk control techniques (see Jeanblanc-Pique and Shiryaev [11], Asmussen and Taksar [2], Radner and Shepp [16], Boyle, Elliott, and Yang [3], H0jgaard and Taksar [8], [9], [10], Paulsen and Gjessing [13], and Taksar and Zhou [18]). In those models the liquid assets of the company are modeled by a Brownian motion with constant drift and diffusion coefficients. The drift term corresponds to the expected (potential) profit per unit time, while the diffusion term is interpreted as risk. The larger the diffusion coefficient the greater the business risk the company takes on. If the company wants to decrease the risk from its business activities, it also faces a decrease in its potential profit. In other words, different business activities in this model correspond to changing simultaneously the drift and the diffusion coefficients of the underlying process. This sets a scene for an optimal stochastic control model where the controls affect not only the drift but also the diffusion part of the dynamic of the system.

Another important feature of our paper is dividend distribution. Dividends are paid from the liquid reserve of the company and distributed to the shareholders.

* Received by the editors December 19. 2000; accepted for publication (in revised form) September 7, 2002; published electronically March 13, 2003.

^Mathematical and Statistical Sciences Department, University of Alberta, Edmonton, AB, T6G2G1 Canada. This author wishes to gratefully acknowledge the financial support and hospitality of the Department of Systems Engineering and Engineering Management at the Chinese University of Hong Kong—where the main part of this work was done—and the Pacific Institute for Mathematical Sciences.

■'"Department of Mathematics, University of Missouri, Columbia, MO 65211 (taksar@math. misgouri.edu). This author is supported by the National Science Foundation grant DMS 9705011.

§ Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong (xyzhou@se.cuhk.edu.hk). This author is supported by the RGC earmarked grant CUHK 4054/98E.

In conclusion, we would like to point out an intricate interplay between the liability and restrictions on the risk control of a financial company. The sheer number of qualitatively different optimal policies, which appears due to different possible relationships between exogenous parameters, shows the multiplicity of different economic environments which a financial company faces depending on the size of the debt and on the size of available business activity.

Acknowledgments. We thank the associate editor and the two reviewers for their careful reading of an earlier version of the paper and for their constructive comments that led to an improved version.

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