Classical and Impulse Stochastic Control on the Optimization of Dividends with Residual Capital at Bankruptcy

Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 2693568, 14 pages https://doi.org/10.1155/2017/2693568 Research Article Classical and Impulse Stochastic Control on the Optimization of Dividends with Residual Capital at Bankruptcy Peimin Chen1 and Bo Li2 1 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China School of Finance, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China 2 Correspondence should be addressed to Peimin Chen; Received 10 October 2016; Accepted 6 February 2017; Published 23 February 2017 Academic Editor: Yong Zhou Copyright © 2017 Peimin Chen and Bo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we consider the optimization problem of dividends for the terminal bankruptcy model, in which some money would be returned to shareholders at the state of terminal bankruptcy, while accounting for the tax rate and transaction cost for dividend payout. Maximization of both expected total discounted dividends before bankruptcy and expected discounted returned money at the state of terminal bankruptcy becomes a mixed classical-impulse stochastic control problem. In order to solve this problem, we reduce it to quasi-variational inequalities with a nonzero boundary condition. We explicitly construct and verify solutions of these inequalities and present the value function together with the optimal policy. 1. Introduction Corporate dividend policy has long engaged the attention of financial economists. The classical paper [1] by Miller and Modigliani provides the valuation formula for an infinite horizon firm under perfect certainty. In a world of perfect capital markets, they show that the dividend policy is irrelevant as a firm can always raise funds to meet the need for continuing operation. However, under a more realistic condition with imperfections such as the presence of financial constraints, information asymmetry, agency costs, taxes, risk exposure under uncertainty, transaction costs, and other frictions, it has been shown that there exists an optimal dividend policy (see [2–5]). Thus, determining the optimal dividend payouts becomes an important issue as it affects firm value. More recent models have focused on the issue of how to set the optimal dividend policy in a dynamic uncertain environment. The valuation model used by Miller-Modigliani in 1961 can be extended to the situation of controllable business activities in a stochastic environment. During the recent decades, there have been increasing interests in applying diffusion models to financial decision problems, especially in (re)insurance modelling (see [6–17]). For most of these models, the liquid assets processes of the corporation contain a Brownian motion with drift and diffusion terms. The drift term corresponds to the expected profit per unit time, and the diffusion term represents risk exposure. By using diffusion models, many kinds of optimal dividend problems, such as in [7, 14–16], are discussed and optimal policies are presented in these papers. Particularly, in some papers (see [14–16, 18]), authors discuss much more practical problems by considering a fixed transaction cost for each dividend payout. In [14], the optimal dividend problem without bankruptcy for insurance firms is considered under the assumptions of constant tax rate and fixed cost for dividend payout. In [15, 16], the author considers the general income process 𝑋𝑡 with drift term 𝜇(𝑋𝑡 ) and diffusion term 𝜎(𝑋𝑡 ) and the case of bankruptcy. Moreover, numerical methods, such as the Runge-Kutta method, are implemented to simulate the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear differential equation. In [19], the author studies the dual risk model with a barrier strategy under the concept of bankruptcy, in which one has a positive probability to continue business despite temporary negative surplus. In [20], the author considers an insurance entity endowed with 2 an initial capital and an income, modelled as a Brownian motion with drift and finds an explicit expression for the value function and for the optimal strategy in the first but not in the second case, where one has to switch to the viscosity ansatz. In [21], the authors suppose that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals and obtain the explicit solutions for value function and optimal strategy. In these papers, the value function is typically assumed to be zero when there is a bankruptcy. But in the real world, some shareholders, especially for preferred shareholders, can get some money back when a terminal bankruptcy occurs. That means for this case the value function is not zero at bankruptcy. Thus, it is very useful and necessary for us to consider this kind of problem. In this paper, we postulate that the amount of money, shareholders can obtain for the terminal bankruptcy, is a positive constant, 𝑎. Moreover, we assume that the liquid assets 𝑋𝑡 follows a process with constant drift and diffusion coefficients. In the model of this paper, as that in [14], the dividend distribution policy is given by a purely discontinuous increasing functional. The net amount of money received by shareholders is 𝑘𝜉𝑖 − 𝐾 for the 𝑖-th dividends, where 𝜉𝑖 is the amount of the dividend payments, 1 − 𝑘 is the tax rate the shareholder pays, and 𝐾 is the fixed cost whenever the dividends are paid out. Further, 𝜏𝑖 represents the moments of dividend payments and 𝜆 is the discount rate. Based on these assumptions, we transform the value function 𝑉(𝑥) into quasi-variational inequalities (QVI) and list out a candidate solution V(𝑥) to QVI with a positive boundary condition V(0) = 𝑎. Subsequently, we show that the value function can be given by V(𝑥) and the optimal policy can be presented based on the solution V(𝑥). A natural question is how to point out whether there is a bankruptcy or not in order to obtain the optimal policy under some conditions. To answer this question, some criteria are provided. A major difficulty in this paper is that the structure of the candidate solution is uncertain since the existing interval of it has unfixed endpoints, which depends on some unknown parameters. This phenomenon does not appear in [14] and other related papers. Enlightened by the derivatives of candidate solutions, we construct the integral 𝐼(𝐶) and then discuss it by several cases for 𝜇, 𝜎, 𝑘, 𝐾, and 𝑎. For the model mentioned above, which is restricted to stay at the bankruptcy state, it is denoted by terminal bankruptcy model as in [22]. The structure of this paper is as follows. In the next section, we provide a rigorous mathematical model for the optimal dividend problem. Then the stochastic contr (...truncated)