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October  2018, 14(4): 1685-1700. doi: 10.3934/jimo.2018027

## Optimal impulse control of a mean-reverting inventory with quadratic costs

 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author: Jinbiao Wu

Received  June 2016 Revised  November 2017 Published  February 2018

Fund Project: The second author is supported by the National Natural Science Foundation of China under Grant 11671404. The third author is supported by the Provincial Natural Science Foundation of Hunan under Grant 2017JJ3405 and Innovation-Driven of Central South University (10900-506010101) and the Yu Ying project of Central South University.

In this paper, we analyze an optimal impulse control problem of a stochastic inventory system whose state follows a mean-reverting Ornstein-Uhlenbeck process. The objective of the management is to keep the inventory level as close as possible to a given target. When the management intervenes in the system, it requires costs consisting of a quadratic form of the system state. Besides, there are running costs associated with the difference between the inventory level and the target. Those costs are also of a quadratic form. The objective of this paper is to find an optimal control of minimizing the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. We solve the problem by using stochastic impulse control theory.

Citation: Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027
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##### References:
Function $D(x)$
Function $D'(x)$
effect of changes in k
 $k$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.1 0.9341 3.3583 4.6138 7.0656 2.4242 2.4518 0.15 0.8622 3.3127 4.6675 7.1500 2.4505 2.4925 0.2 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.25 0.6962 3.2035 4.7994 7.3479 2.5073 2.5485 0.3 0.6005 3.1385 4.8793 7.4636 2.5379 2.5843 $\sigma$=1.2, $\rho$=4.0, $\lambda$=0.06, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
 $k$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.1 0.9341 3.3583 4.6138 7.0656 2.4242 2.4518 0.15 0.8622 3.3127 4.6675 7.1500 2.4505 2.4925 0.2 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.25 0.6962 3.2035 4.7994 7.3479 2.5073 2.5485 0.3 0.6005 3.1385 4.8793 7.4636 2.5379 2.5843 $\sigma$=1.2, $\rho$=4.0, $\lambda$=0.06, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
effect of changes in $\lambda$
 $\lambda$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.01 0.8506 3.2915 4.6914 7.1665 2.4409 2.4751 0.03 0.8239 3.2797 4.7063 7.1917 2.4558 2.4908 0.05 0.7968 3.2676 4.7214 7.2281 2.4708 2.5067 0.06 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.07 0.7695 3.2551 4.7368 7.2595 2.4856 2.5227 $\sigma$=1.2, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
 $\lambda$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.01 0.8506 3.2915 4.6914 7.1665 2.4409 2.4751 0.03 0.8239 3.2797 4.7063 7.1917 2.4558 2.4908 0.05 0.7968 3.2676 4.7214 7.2281 2.4708 2.5067 0.06 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.07 0.7695 3.2551 4.7368 7.2595 2.4856 2.5227 $\sigma$=1.2, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
effect of changes in $\sigma$
 $\sigma$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 1.1 0.9030 3.2959 4.6989 7.1259 2.3929 2.4270 1.2 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 1.3 0.6661 3.2265 4.7595 7.3587 2.5604 2.5992 1.4 0.5517 3.1913 4.7901 7.4710 2.6396 2.6800 1.5 0.4397 3.1559 4.8207 7.5807 2.7162 2.7600 $\lambda$=0.06, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
 $\sigma$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 1.1 0.9030 3.2959 4.6989 7.1259 2.3929 2.4270 1.2 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 1.3 0.6661 3.2265 4.7595 7.3587 2.5604 2.5992 1.4 0.5517 3.1913 4.7901 7.4710 2.6396 2.6800 1.5 0.4397 3.1559 4.8207 7.5807 2.7162 2.7600 $\lambda$=0.06, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
effect of changes in $\rho$
 $\rho$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 3.5 0.2915 2.7717 4.2182 6.7351 2.4802 2.5169 3.8 0.5865 3.0655 4.5248 7.0403 2.4790 2.5155 4.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 4.5 1.2750 3.7511 5.2399 7.7523 2.4761 2.5124 $\lambda$=0.06, $\sigma$=1.2, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
 $\rho$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 3.5 0.2915 2.7717 4.2182 6.7351 2.4802 2.5169 3.8 0.5865 3.0655 4.5248 7.0403 2.4790 2.5155 4.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 4.5 1.2750 3.7511 5.2399 7.7523 2.4761 2.5124 $\lambda$=0.06, $\sigma$=1.2, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
effect of changes in $k_{21}$, $k_{22}$, $k_{11}$, $k_{12}$, $k_{01}$, and $k_{02}$
 $k_{21}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.04 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.1 0.7229 3.1639 4.7291 7.2495 2.4410 2.5103 0.2 0.6298 3.0140 4.7595 7.2579 2.3842 2.5041 0.4 0.4672 2.7527 4.8207 7.2650 2.2855 2.4946 $k_{22}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.04 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.1 0.7669 3.2333 4.8442 7.3653 2.4664 2.5211 0.2 0.7434 3.1933 5.0523 7.5849 2.4499 2.5326 0.4 0.7090 3.1358 5.5454 8.1028 2.4268 2.5574 $k_{11}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 2.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 2.5 0.6920 3.1539 4.7500 7.2557 2.4619 2.5057 3.0 0.6008 3.0469 4.7688 7.2666 2.4461 2.4978 4.0 0.4177 2.8341 4.8010 7.2854 2.4164 2.4844 $k_{12}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 2.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 2.5 0.7719 3.2418 4.8419 7.3389 2.4699 2.4970 3.0 0.7616 3.2242 4.9540 7.4341 2.4626 2.4801 4.0 0.7440 3.1943 5.1769 7.6251 2.4503 2.4482 $k_{01}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 5.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 6.0 0.6670 3.3015 4.7457 7.2532 2.6345 2.5075 7.0 0.5606 3.3353 4.7602 7.2616 2.7747 2.5014 8.0 0.4619 3.3644 4.7731 7.2691 2.9025 2.4960 $k_{02}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 5.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 6.0 0.7743 3.2460 4.6889 7.3622 2.4717 2.6733 7.0 0.7665 3.2326 4.6551 7.4707 2.4661 2.8156 8.0 0.7596 3.2207 4.6260 7.5713 2.4611 2.9453 The default parameters in the calculations are $\rho$=4.0, $\lambda$=0.06, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
 $k_{21}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.04 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.1 0.7229 3.1639 4.7291 7.2495 2.4410 2.5103 0.2 0.6298 3.0140 4.7595 7.2579 2.3842 2.5041 0.4 0.4672 2.7527 4.8207 7.2650 2.2855 2.4946 $k_{22}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 0.04 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 0.1 0.7669 3.2333 4.8442 7.3653 2.4664 2.5211 0.2 0.7434 3.1933 5.0523 7.5849 2.4499 2.5326 0.4 0.7090 3.1358 5.5454 8.1028 2.4268 2.5574 $k_{11}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 2.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 2.5 0.6920 3.1539 4.7500 7.2557 2.4619 2.5057 3.0 0.6008 3.0469 4.7688 7.2666 2.4461 2.4978 4.0 0.4177 2.8341 4.8010 7.2854 2.4164 2.4844 $k_{12}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 2.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 2.5 0.7719 3.2418 4.8419 7.3389 2.4699 2.4970 3.0 0.7616 3.2242 4.9540 7.4341 2.4626 2.4801 4.0 0.7440 3.1943 5.1769 7.6251 2.4503 2.4482 $k_{01}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 5.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 6.0 0.6670 3.3015 4.7457 7.2532 2.6345 2.5075 7.0 0.5606 3.3353 4.7602 7.2616 2.7747 2.5014 8.0 0.4619 3.3644 4.7731 7.2691 2.9025 2.4960 $k_{02}$ $a$ $\alpha$ $\beta$ b $\alpha-a$ $b-\beta$ 5.0 0.7832 3.2614 4.7291 7.2437 2.4782 2.5146 6.0 0.7743 3.2460 4.6889 7.3622 2.4717 2.6733 7.0 0.7665 3.2326 4.6551 7.4707 2.4661 2.8156 8.0 0.7596 3.2207 4.6260 7.5713 2.4611 2.9453 The default parameters in the calculations are $\rho$=4.0, $\lambda$=0.06, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
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