The Optimal Dividend Problem in Dual Model with Capital Injections by Stochastic Interest Rates
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摘要: 在红利有界的条件下, 讨论了复合二项对偶模型中带比例交易费再注资且分红贴现利率随机变化的最优分红问题; 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散的HJB方程的唯一解, 得到了最优分红策略和最优值函数的计算方法; 根据分红策略的一些性质, 得到了该最优值函数的可无限逼近的上界和下界，并采用了Bellman递归算法得到最优值函数和最优分红策略的数值解，从而得到最优分红算法.数值实例结果表明:该最优分红策略是有效的.这为公司的决策者在兼顾公司正常运营和股东利益而进行红利决策时提供了理论依据.Abstract: Under the dividend bounded, the problem of optimal dividend payment in compound binomial dual model with proportional transaction cost capital injections and stochastic interest rates was discussed. It was proved with the fixed-point principle of contraction mapping that the optimal value function of this optimal dividend problem was the unique solution to a discrete Hamilton-Jacobi-Bellman(HJB) equation. The algorithm was obtained for the optimal dividend strategy and the optimal value function. According to some properties of the dividend strategy, the upper and lower bounds of the optimal value function were derived, and the numerical solutions to the optimal value function and the optimal dividend strategy was obtained with the Bellman recursive algorithm, and the optimal dividend algorithm was obtained. The numerical result shows that the optimal dividend strategy is effective. A theoretical basis for the decision-maker to make dividend policy in consideration of the normal operation of the company and interests of shareholders is provided.
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Key words:
- optimal dividend /
- stochastic interest rates /
- capital injections /
- HJB equation
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表 1 最优分红策略φi*(u) (i=1, 2, 3, C=5)
Table 1. The optimal dividend strategy φi*(u) (i=1, 2, 3, C=5)
u β=0 β=0.2 β=0.4 β=0.6 β=1.2 不注资 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 0, 1, …, 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 12 2 2 2 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 13 3 3 3 3 3 3 2 1 1 0 0 0 0 0 0 0 0 0 14 4 4 4 4 4 4 3 2 2 0 0 0 0 0 0 0 0 0 15 5 5 5 5 5 5 4 3 3 0 0 0 0 0 0 0 0 0 16 5 5 5 5 5 5 5 4 4 1 1 1 0 0 0 0 0 0 17 5 5 5 5 5 5 5 5 5 2 2 2 0 0 0 0 0 0 18 5 5 5 5 5 5 5 5 5 3 3 3 0 0 0 0 0 0 19 5 5 5 5 5 5 5 5 5 4 4 4 0 0 0 0 0 0 20 5 5 5 5 5 5 5 5 5 5 5 5 0 0 0 0 0 0 21 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 22 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 23 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 24 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 25, 26, … 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 
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表 2 最优值函数
Table 2. The optimal value function
u β=0 β=0.2 β=1.2 不注资 V1*(u) V2*(u) V3*(u) V1*(u) V2*(u) V3*(u) V1*(u) V2*(u) V3*(u) W1*(u) W2*(u) W3*(u) 0 34.650 1 35.061 4 35.033 0 29.566 9 29.938 8 29.912 0 8.898 5 9.126 3 9.102 4 10.566 7 10.673 5 10.667 2 1 35.650 1 36.061 4 36.033 0 30.766 9 31.138 8 31.112 0 11.098 5 11.326 3 11.302 4 11.740 8 11.859 4 11.852 5 2 36.650 1 37.061 4 37.033 0 31.966 9 32.338 8 32.312 0 13.298 5 13.526 3 13.502 4 13.045 4 13.177 2 13.169 4 3 37.650 1 38.061 4 38.033 0 33.166 9 33.538 8 33.512 0 15.498 5 15.726 3 15.702 4 14.494 9 14.641 3 14.632 7 4 38.650 1 39.061 4 39.033 0 34.366 9 34.738 8 34.712 0 17.698 5 17.926 3 17.902 4 16.105 4 16.268 1 16.258 5 5 39.650 1 40.061 4 40.033 0 35.566 9 35.938 8 35.912 0 19.898 5 20.126 3 20.102 4 20.153 1 20.357 0 20.344 9 6 40.563 5 40.981 1 40.953 3 36.611 9 36.990 9 36.964 8 21.621 3 21.861 2 21.838 3 21.514 1 21.731 7 21.718 8 7 41.471 9 41.896 0 41.868 7 37.645 3 38.031 5 38.006 1 23.301 5 23.553 2 23.531 4 22.928 8 23.160 6 23.146 9 8 42.374 9 42.805 5 42.778 7 38.665 7 39.059 2 39.034 4 24.934 2 25.197 8 25.176 8 24.392 3 24.638 7 24.624 2 9 43.271 8 43.708 9 43.682 6 39.671 9 40.072 7 40.048 5 26.514 1 26.789 7 26.769 4 25.897 9 26.159 4 26.144 0 10 44.161 9 44.605 7 44.579 8 40.662 1 41.070 4 41.046 6 28.035 4 28.322 8 28.303 1 27.919 5 28.201 3 28.184 9 11 45.026 6 45.477 8 45.452 1 41.602 2 42.018 6 41.995 0 29.391 2 29.691 0 29.671 3 29.159 1 29.453 2 29.436 1 12 45.888 8 46.347 3 46.321 8 42.532 9 42.957 1 42.933 7 30.704 5 31.016 4 30.996 6 30.402 4 30.708 8 30.691 0 13 46.748 7 47.214 4 47.189 1 43.453 8 43.885 9 43.862 6 31.974 3 32.298 0 32.278 1 31.643 1 31.961 7 31.943 4 14 47.606 4 48.079 3 48.054 2 44.365 1 44.804 8 44.781 8 33.199 6 33.534 7 33.514 7 32.874 3 33.204 9 33.186 1 15 48.462 3 48.942 3 48.917 3 45.266 9 45.714 1 45.691 2 34.380 0 34.726 1 34.705 8 34.191 7 34.535 2 34.515 8 16 49.341 0 49.827 5 49.802 9 46.164 5 46.618 4 46.596 0 35.493 8 35.850 4 35.829 9 35.272 3 35.626 3 35.606 4 17 50.210 9 50.704 0 50.679 8 47.055 5 47.516 2 47.494 1 36.577 4 36.944 3 36.923 6 36.341 4 36.705 5 36.685 3 18 51.071 1 51.570 8 51.546 9 47.939 0 48.406 6 48.384 8 37.631 9 38.008 7 37.987 7 37.395 8 37.770 0 37.749 4 19 51.920 3 52.426 9 52.403 2 48.814 2 49.289 0 49.267 4 38.658 4 39.044 8 39.023 6 38.432 8 38.816 7 38.795 8 20 52.757 5 53.271 1 53.247 6 49.680 3 50.162 3 50.140 8 39.658 4 40.054 0 40.032 6 39.471 9 39.865 5 39.844 3 21 53.553 2 54.074 4 54.050 7 50.524 1 51.014 3 50.992 6 40.629 3 41.033 7 41.012 2 40.435 0 40.837 4 40.816 1 22 54.344 3 54.873 2 54.849 3 51.359 2 51.857 5 51.835 6 41.582 6 41.995 6 41.974 0 41.387 3 41.798 3 41.776 8 23 55.130 8 55.667 3 55.643 1 52.185 7 52.692 0 52.669 8 42.519 7 42.941 1 42.919 4 42.328 0 42.747 4 42.725 8 24 55.912 3 56.456 4 56.432 0 53.003 5 53.517 7 53.495 4 43.442 4 43.871 9 43.850 2 43.256 8 43.684 4 43.662 8 25 56.688 7 57.240 3 57.215 7 53.812 9 54.335 0 54.312 3 44.352 3 44.789 6 44.767 9 44.178 5 44.614 0 44.592 5 26 57.464 8 58.023 8 57.999 1 54.614 9 55.144 7 55.121 9 45.255 6 45.699 7 45.678 4 45.082 5 45.524 8 45.503 6 27 58.231 4 58.797 9 58.772 9 55.408 8 55.946 3 55.923 3 46.152 9 46.603 9 46.582 9 45.980 5 46.429 6 46.408 8 28 58.988 2 59.562 1 59.536 9 56.194 2 56.739 4 56.716 1 47.043 5 47.501 4 47.480 7 46.871 8 47.328 0 47.307 4 29 59.735 0 60.316 3 60.290 7 56.970 8 57.523 7 57.500 0 47.926 5 48.391 7 48.371 1 47.755 7 48.219 2 48.198 7 30 60.471 5 61.060 1 61.034 2 57.738 3 58.298 8 58.274 8 48.801 3 49.273 9 49.253 4 48.631 4 49.102 3 49.081 9
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