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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

Yanqing Hu , Zaiming Liu and  Jinbiao Wu ,

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

* Corresponding author: Jinbiao Wu

Received  June 2016 Revised  November 2017 Published  October 2018 Early access  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.

Keywords: Stochastic impulse control, inventory control, mean-reverting Ornstein-Uhlenbeck process, quasi-variational inequalities, quadratic costs.
Mathematics Subject Classification: Primary: 93E20, 90B05; Secondary: 49J40.
Citation: Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027
References:
[1]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Gaunthier-Villars, paris, 1984.

[2]

G. Bertola, W. J. Runggaldier and K. Yasuda, On classical and restricted impulse stochastic control for the exchange rate, Applied Mathematics and Optimization, 74 (2016), 423-454.  doi: 10.1007/s00245-015-9320-6.

[3]

O. Bouasker, N. Letifi and J. L. Prigent, Optimal funding and hiring/firing policies with mean reverting demand, Economic Modeling, 58 (2016), 569-579.  doi: 10.1016/j.econmod.2015.11.015.

[4]

K. A. Brekke and B. Φksendal, A verification theorem for combined stochastic control and impulse control, Stochastic Analysis and Related Topics, 42 (1998), 211-220. 

[5]

A. Cadenillas and F. Zapatero, Optimal central bank intervention in the foreign exchange market, Journal of Economic Theory, 87 (1999), 218-242.  doi: 10.1006/jeth.1999.2523.

[6]

A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Mathematical Finance, 10 (2000), 141-156.  doi: 10.1111/1467-9965.00086.

[7]

A. Cadenillas, Consumption-investment problems with transaction costs: Survey and open problems, Mathematical Methods of Operations Research, 51 (2000), 43-68.  doi: 10.1007/s001860050002.

[8]

A. Cadenillas, P. Lakner and M. Pinedo, Optimal control of a mean-reverting inventory, Operations Research, 58 (2010), 1697-1710.  doi: 10.1287/opre.1100.0835.

[9]

G. M. Constantinides and S. F. Richard, Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time, Operations Research, 26 (1978), 620-636.  doi: 10.1287/opre.26.4.620.

[10]

H. Feng, Q. Wu, K. Muthuraman and V. Deshpande, Replenishment policies for multi-product stochastic inventory systems with correlated demand and joint-replenishment costs, Production and Operations Management, 24 (2015), 647-664.  doi: 10.1111/poms.12290.

[11]

T. Gescheidt, V. I. Losada and K. Mensikova, Impulse control disorders in patients with young-onset Parkinson's disease: a cross-sectional study seeking associated factors, Basal Ganglia, 6 (2016), 197-205.  doi: 10.1016/j.baga.2016.09.001.

[12]

J. M. Harrison, T. M. Sellke and A. J. Taylor, Impulse control of brownian motion, Mathematics of Operations Research, 8 (1983), 454-466.  doi: 10.1287/moor.8.3.454.

[13]

A. Madhavan and S. Smidt, An analysis of changes in specialist inventories and quotations, Journal of Finance, 48 (1993), 1595-1628.  doi: 10.1111/j.1540-6261.1993.tb05122.x.

[14]

D. W. Miao, X. C. Lin and W. L. Chao, Option pricing under jump-diffusion model with mean-reverting bivariate jumps, Operations Research Letters, 42 (2014), 27-33.  doi: 10.1016/j.orl.2013.11.004.

[15]

M. Ohnishi and M. Tsujimura, An impulse control of a geometric brownian motion with quadratic costs, European Journal of Operational Research, 168 (2006), 311-321.  doi: 10.1016/j.ejor.2004.07.006.

[16]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition. Universitext. Springer, Berlin, 2007.

[17]

M. Ormeci, J. G. Dai and J. Vande Vate, Impulse control of Brownian motion: The constrained average cost case, Operations Research, 56 (2008), 618-629.  doi: 10.1287/opre.1060.0380.

[18]

X. J. Pan and S. D. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628.  doi: 10.1080/00207543.2014.961201.

[19]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Cambridge University Press, 2000.

[20]

L. Vela, J. C. M. Castrillo and P. J. Garcia-Ruiz, The high prevalence of impulse control behaviors in patients with early-onset Parkinson's disease: A cross-sectional multicenter study, Journal of Neurological Sciences, 368 (2016), 150-154.  doi: 10.1016/j.jns.2016.07.003.

[21]

A. Weerasinghe and C. Zhu, Optimal inventory control with path-dependent cost criteria, Stochastic Processes and their Applications, 126 (2016), 1585-1621.  doi: 10.1016/j.spa.2015.11.014.

[22]

D. Yao, X. Chao and J. Wu, Optimal control policy for a Brownian inventory system with concave ordering cost, Journal of Applied Probability, 52 (2015), 909-925.  doi: 10.1017/S0021900200112987.

[23]

S. Young, Transaction Cost Economics Springer Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-28036-8_221.

show all references

References:
[1]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Gaunthier-Villars, paris, 1984.

[2]

G. Bertola, W. J. Runggaldier and K. Yasuda, On classical and restricted impulse stochastic control for the exchange rate, Applied Mathematics and Optimization, 74 (2016), 423-454.  doi: 10.1007/s00245-015-9320-6.

[3]

O. Bouasker, N. Letifi and J. L. Prigent, Optimal funding and hiring/firing policies with mean reverting demand, Economic Modeling, 58 (2016), 569-579.  doi: 10.1016/j.econmod.2015.11.015.

[4]

K. A. Brekke and B. Φksendal, A verification theorem for combined stochastic control and impulse control, Stochastic Analysis and Related Topics, 42 (1998), 211-220. 

[5]

A. Cadenillas and F. Zapatero, Optimal central bank intervention in the foreign exchange market, Journal of Economic Theory, 87 (1999), 218-242.  doi: 10.1006/jeth.1999.2523.

[6]

A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Mathematical Finance, 10 (2000), 141-156.  doi: 10.1111/1467-9965.00086.

[7]

A. Cadenillas, Consumption-investment problems with transaction costs: Survey and open problems, Mathematical Methods of Operations Research, 51 (2000), 43-68.  doi: 10.1007/s001860050002.

[8]

A. Cadenillas, P. Lakner and M. Pinedo, Optimal control of a mean-reverting inventory, Operations Research, 58 (2010), 1697-1710.  doi: 10.1287/opre.1100.0835.

[9]

G. M. Constantinides and S. F. Richard, Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time, Operations Research, 26 (1978), 620-636.  doi: 10.1287/opre.26.4.620.

[10]

H. Feng, Q. Wu, K. Muthuraman and V. Deshpande, Replenishment policies for multi-product stochastic inventory systems with correlated demand and joint-replenishment costs, Production and Operations Management, 24 (2015), 647-664.  doi: 10.1111/poms.12290.

[11]

T. Gescheidt, V. I. Losada and K. Mensikova, Impulse control disorders in patients with young-onset Parkinson's disease: a cross-sectional study seeking associated factors, Basal Ganglia, 6 (2016), 197-205.  doi: 10.1016/j.baga.2016.09.001.

[12]

J. M. Harrison, T. M. Sellke and A. J. Taylor, Impulse control of brownian motion, Mathematics of Operations Research, 8 (1983), 454-466.  doi: 10.1287/moor.8.3.454.

[13]

A. Madhavan and S. Smidt, An analysis of changes in specialist inventories and quotations, Journal of Finance, 48 (1993), 1595-1628.  doi: 10.1111/j.1540-6261.1993.tb05122.x.

[14]

D. W. Miao, X. C. Lin and W. L. Chao, Option pricing under jump-diffusion model with mean-reverting bivariate jumps, Operations Research Letters, 42 (2014), 27-33.  doi: 10.1016/j.orl.2013.11.004.

[15]

M. Ohnishi and M. Tsujimura, An impulse control of a geometric brownian motion with quadratic costs, European Journal of Operational Research, 168 (2006), 311-321.  doi: 10.1016/j.ejor.2004.07.006.

[16]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition. Universitext. Springer, Berlin, 2007.

[17]

M. Ormeci, J. G. Dai and J. Vande Vate, Impulse control of Brownian motion: The constrained average cost case, Operations Research, 56 (2008), 618-629.  doi: 10.1287/opre.1060.0380.

[18]

X. J. Pan and S. D. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628.  doi: 10.1080/00207543.2014.961201.

[19]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Cambridge University Press, 2000.

[20]

L. Vela, J. C. M. Castrillo and P. J. Garcia-Ruiz, The high prevalence of impulse control behaviors in patients with early-onset Parkinson's disease: A cross-sectional multicenter study, Journal of Neurological Sciences, 368 (2016), 150-154.  doi: 10.1016/j.jns.2016.07.003.

[21]

A. Weerasinghe and C. Zhu, Optimal inventory control with path-dependent cost criteria, Stochastic Processes and their Applications, 126 (2016), 1585-1621.  doi: 10.1016/j.spa.2015.11.014.

[22]

D. Yao, X. Chao and J. Wu, Optimal control policy for a Brownian inventory system with concave ordering cost, Journal of Applied Probability, 52 (2015), 909-925.  doi: 10.1017/S0021900200112987.

[23]

S. Young, Transaction Cost Economics Springer Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-28036-8_221.

Figure 1.  Function $D(x)$
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Figure 2.  Function $D'(x)$
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Table 1.  effect of changes in k
$k$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.10.93413.35834.61387.06562.42422.4518
0.150.86223.31274.66757.15002.45052.4925
0.20.78323.26144.72917.24372.47822.5146
0.250.69623.20354.79947.34792.50732.5485
0.30.60053.13854.87937.46362.53792.5843
$\sigma$=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.
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Download as excel

$k$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.10.93413.35834.61387.06562.42422.4518
0.150.86223.31274.66757.15002.45052.4925
0.20.78323.26144.72917.24372.47822.5146
0.250.69623.20354.79947.34792.50732.5485
0.30.60053.13854.87937.46362.53792.5843
$\sigma$=1.2, $\rho$=4.0, $\lambda$=0.06, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
Table 2.  effect of changes in $\lambda$
$\lambda$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.010.85063.29154.69147.16652.44092.4751
0.030.82393.27974.70637.19172.45582.4908
0.050.79683.26764.72147.22812.47082.5067
0.060.78323.26144.72917.24372.47822.5146
0.070.76953.25514.73687.25952.48562.5227
$\sigma$=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.
Table Options

Download as excel

$\lambda$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.010.85063.29154.69147.16652.44092.4751
0.030.82393.27974.70637.19172.45582.4908
0.050.79683.26764.72147.22812.47082.5067
0.060.78323.26144.72917.24372.47822.5146
0.070.76953.25514.73687.25952.48562.5227
$\sigma$=1.2, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
Table 3.  effect of changes in $\sigma$
$\sigma$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
1.10.90303.29594.69897.12592.39292.4270
1.20.78323.26144.72917.24372.47822.5146
1.30.66613.22654.75957.35872.56042.5992
1.40.55173.19134.79017.47102.63962.6800
1.50.43973.15594.82077.58072.71622.7600
$\lambda$=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.
Table Options

Download as excel

$\sigma$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
1.10.90303.29594.69897.12592.39292.4270
1.20.78323.26144.72917.24372.47822.5146
1.30.66613.22654.75957.35872.56042.5992
1.40.55173.19134.79017.47102.63962.6800
1.50.43973.15594.82077.58072.71622.7600
$\lambda$=0.06, $\rho$=4.0, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
Table 4.  effect of changes in $\rho$
$\rho$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
3.50.29152.77174.21826.73512.48022.5169
3.80.58653.06554.52487.04032.47902.5155
4.00.78323.26144.72917.24372.47822.5146
4.51.27503.75115.23997.75232.47612.5124
$\lambda$=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.
Table Options

Download as excel

$\rho$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
3.50.29152.77174.21826.73512.48022.5169
3.80.58653.06554.52487.04032.47902.5155
4.00.78323.26144.72917.24372.47822.5146
4.51.27503.75115.23997.75232.47612.5124
$\lambda$=0.06, $\sigma$=1.2, $k$=0.2, $k_{01}$=5.0, $k_{02}$=5.0, $k_{21}$=0.04, $k_{22}$=0.04, $k_{11}$=2.0, $k_{12}$=2.0.
Table 5.  effect of changes in $k_{21}$, $k_{22}$, $k_{11}$, $k_{12}$, $k_{01}$, and $k_{02}$
$k_{21}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.040.78323.26144.72917.24372.47822.5146
0.10.72293.16394.72917.24952.44102.5103
0.20.62983.01404.75957.25792.38422.5041
0.40.46722.75274.82077.26502.28552.4946
$k_{22}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.040.78323.26144.72917.24372.47822.5146
0.10.76693.23334.84427.36532.46642.5211
0.20.74343.19335.05237.58492.44992.5326
0.40.70903.13585.54548.10282.42682.5574
$k_{11}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
2.00.78323.26144.72917.24372.47822.5146
2.50.69203.15394.75007.25572.46192.5057
3.00.60083.04694.76887.26662.44612.4978
4.00.41772.83414.80107.28542.41642.4844
$k_{12}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
2.00.78323.26144.72917.24372.47822.5146
2.50.77193.24184.84197.33892.46992.4970
3.00.76163.22424.95407.43412.46262.4801
4.00.74403.19435.17697.62512.45032.4482
$k_{01}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
5.00.78323.26144.72917.24372.47822.5146
6.00.66703.30154.74577.25322.63452.5075
7.00.56063.33534.76027.26162.77472.5014
8.00.46193.36444.77317.26912.90252.4960
$k_{02}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
5.00.78323.26144.72917.24372.47822.5146
6.00.77433.24604.68897.36222.47172.6733
7.00.76653.23264.65517.47072.46612.8156
8.00.75963.22074.62607.57132.46112.9453
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.
Table Options

Download as excel

$k_{21}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.040.78323.26144.72917.24372.47822.5146
0.10.72293.16394.72917.24952.44102.5103
0.20.62983.01404.75957.25792.38422.5041
0.40.46722.75274.82077.26502.28552.4946
$k_{22}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
0.040.78323.26144.72917.24372.47822.5146
0.10.76693.23334.84427.36532.46642.5211
0.20.74343.19335.05237.58492.44992.5326
0.40.70903.13585.54548.10282.42682.5574
$k_{11}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
2.00.78323.26144.72917.24372.47822.5146
2.50.69203.15394.75007.25572.46192.5057
3.00.60083.04694.76887.26662.44612.4978
4.00.41772.83414.80107.28542.41642.4844
$k_{12}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
2.00.78323.26144.72917.24372.47822.5146
2.50.77193.24184.84197.33892.46992.4970
3.00.76163.22424.95407.43412.46262.4801
4.00.74403.19435.17697.62512.45032.4482
$k_{01}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
5.00.78323.26144.72917.24372.47822.5146
6.00.66703.30154.74577.25322.63452.5075
7.00.56063.33534.76027.26162.77472.5014
8.00.46193.36444.77317.26912.90252.4960
$k_{02}$ $a$ $\alpha$ $\beta$b $\alpha-a$$ b-\beta$
5.00.78323.26144.72917.24372.47822.5146
6.00.77433.24604.68897.36222.47172.6733
7.00.76653.23264.65517.47072.46612.8156
8.00.75963.22074.62607.57132.46112.9453
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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