November  2022, 42(11): 5437-5451. doi: 10.3934/dcds.2022106

The general maximum principle for stochastic control problems with singular controls

1. 

Zhongtai Securities Institute for Financial Study, Shandong University, Jinan Shandong 250100, China

2. 

School of Mathematics and Zhongtai Securities Institute for Financial Study, Shandong University, Jinan Shandong 250100, China

*Corresponding author: Zhen Wu

Received  December 2021 Revised  April 2022 Published  November 2022 Early access  August 2022

Fund Project: This work is supported jointly by the Natural Science Foundation of China [grant number 11831010] and [grant number 61961160732], the Natural Science Foundation of Shandong Province [grant number ZR2019ZD42] and the Taishan Scholars Climbing Program of Shandong [grant number TSPD20210302]

We consider a stochastic control problem with an absolutely continuous control and a singular control. The control domain of the two kinds of admissible controls need not to be convex. We obtain two maximum principles respectively by applying spike variation to these two kinds of optimal controls.

Citation: Yuanzhuo Song, Zhen Wu. The general maximum principle for stochastic control problems with singular controls. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5437-5451. doi: 10.3934/dcds.2022106
References:
[1]

S. Bahlali and A. Chala, The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients, Random Oper. Stoch. Equ., 13 (2005), 1-10.  doi: 10.1515/1569397053300919.

[2]

S. Bahlali and B. Mezerdi, A general stochastic maximum principle for singular control problems, Electron. J. Probab., 10 (2005), 988-1004.  doi: 10.1214/EJP.v10-271.

[3]

A. Cadenillas and U. G. Haussmann, The stochastic maximum principle for a singular control problem, Stoch. Stoch. Rep., 49 (1994), 211-237.  doi: 10.1080/17442509408833921.

[4]

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

[5]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[6]

S. He, J. Wang and J. Yan, Semimartingale Theory and Stochastic Calculus, Routledge, Abingdon, 1992.

[7]

M. Jeanblanc-Picqué, Impulse control method and exchange rate, Math. Finance, 3 (1993), 161-177. 

[8]

R. Korn, Some applications of impulse control in mathematical finance, Math. Methods Oper. Res., 50 (1999), 493-518.  doi: 10.1007/s001860050083.

[9]

B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Springer Science & Business Media, Dordrecht, 2003. doi: 10.1007/978-1-4615-0095-7.

[10]

B. Oksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM J. Control Optim., 40 (2002), 1765-1790.  doi: 10.1137/S0363012900376013.

[11]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[12]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794.

[13]

Z. Wu and F. Zhang, Stochastic maximum principle for optimal control problems of forward-backward systems involving impulse controls, IEEE Trans. Autom. Control, 56 (2011), 1401-1406.  doi: 10.1109/TAC.2011.2114990.

show all references

References:
[1]

S. Bahlali and A. Chala, The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients, Random Oper. Stoch. Equ., 13 (2005), 1-10.  doi: 10.1515/1569397053300919.

[2]

S. Bahlali and B. Mezerdi, A general stochastic maximum principle for singular control problems, Electron. J. Probab., 10 (2005), 988-1004.  doi: 10.1214/EJP.v10-271.

[3]

A. Cadenillas and U. G. Haussmann, The stochastic maximum principle for a singular control problem, Stoch. Stoch. Rep., 49 (1994), 211-237.  doi: 10.1080/17442509408833921.

[4]

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

[5]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[6]

S. He, J. Wang and J. Yan, Semimartingale Theory and Stochastic Calculus, Routledge, Abingdon, 1992.

[7]

M. Jeanblanc-Picqué, Impulse control method and exchange rate, Math. Finance, 3 (1993), 161-177. 

[8]

R. Korn, Some applications of impulse control in mathematical finance, Math. Methods Oper. Res., 50 (1999), 493-518.  doi: 10.1007/s001860050083.

[9]

B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Springer Science & Business Media, Dordrecht, 2003. doi: 10.1007/978-1-4615-0095-7.

[10]

B. Oksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM J. Control Optim., 40 (2002), 1765-1790.  doi: 10.1137/S0363012900376013.

[11]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[12]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794.

[13]

Z. Wu and F. Zhang, Stochastic maximum principle for optimal control problems of forward-backward systems involving impulse controls, IEEE Trans. Autom. Control, 56 (2011), 1401-1406.  doi: 10.1109/TAC.2011.2114990.

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