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December  2020, 10(4): 855-875. doi: 10.3934/mcrf.2020022

Stochastic impulse control Problem with state and time dependent cost functions

Brahim El Asri 1, and  Sehail Mazid 2,

1. 

Equipe. Aide à la decision, Université Ibn Zohr, ENSA, B.P. 1136, Agadir, Maroc
2. 

Department of Mathematical Sciences, Norwegian University of Sciences and Technology, Trondheim, 7491, Norway

Received  August 2019 Revised  November 2019 Published  December 2020 Early access  March 2020

We consider stochastic impulse control problems when the impulses cost functions depend on $ t $ and $ x $. We use the approximation scheme and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems.

Keywords: Stochastic impulse control, quasi-variational inequality, viscosity solution.
Mathematics Subject Classification: 93E20, 35Q93, 35D40.
Citation: Brahim El Asri, Sehail Mazid. Stochastic impulse control Problem with state and time dependent cost functions. Mathematical Control & Related Fields, 2020, 10 (4) : 855-875. doi: 10.3934/mcrf.2020022
References:
[1]

L. H. Alvarez, Stochastic forest stand value and optimal timber harvesting, SIAM J. Control Optim., 42 (2004), 1972–1993 (electronic). doi: 10.1137/S0363012901393456.  Google Scholar

[2]

L. H. Alvarez, A class of solvable impulse control problems, Applied Mathematics and Optimization, 49 (2004), 265-295.  doi: 10.1007/s00245-004-0792-z.  Google Scholar

[3]

L. H. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732.  doi: 10.1137/060659375.  Google Scholar

[4]

P. Azimzadeh, Zero-sum stochastic differential game with impulses, precommitment and unrestricted cost functions, Applied Math. and Optim, 79 (2019), 483-514.  doi: 10.1007/s00245-017-9445-x.  Google Scholar

[5]

G. Barles and C. Imbert, Second order elliptic integro-differential Equations: Viscosity solutions's theory revisited., Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[6]

C. Belak, S. Christensen and F. T. Seifried, A general verification result for stochastic impulse control problems, SIAM J. Control Optim., 55 (2017), 627-649.  doi: 10.1137/16M1082822.  Google Scholar

[7]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, , Gauthier-Villars, Montrouge, 1984.  Google Scholar

[8]

B. Bouchard, A stochastic target formulation for optimal switching problems in finite horizon, Stochastics, 81 (2009), 171-197.  doi: 10.1080/17442500802327360.  Google Scholar

[9]

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.  Google Scholar

[10]

Y-S. A. Chen and X. Guo, Impulse control of multidimensional jump diffusions in finite time horison, SIAM J. Control Optim., 51 (2013), 2638-2663.  doi: 10.1137/110854205.  Google Scholar

[11]

M. Crandall, H. Ishii and P. L. Lions, Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

J. Dugundji, Topolgy, Boston: Allyn and Bacon, US, 1966.  Google Scholar

[13]

B. El Asri, Deterministic minimax impulse control in finite horizon: The viscosity solution approach., ESAIM: Control Optim. Calc. Var., 19 (2013), 63-77.  doi: 10.1051/cocv/2011200.  Google Scholar

[14]

B. El Asri, The value of a minimax problem involving impulse control, Journal of Dynamics and Games, 6 (20419), 1-17.  doi: 10.3934/jdg.2019001.  Google Scholar

[15]

B. El Asri and S. Mazid, Zero-sum stochastic differential game in finite horizon involving Impulse controls, Applied Mathematics and Optimization, 2018. doi: 10.1007/s00245-018-9529-2.  Google Scholar

[16]

B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, Journal of Mathematical Analysis and Applications, 474 (2019), 793-813.  doi: 10.1016/j.jmaa.2019.01.040.  Google Scholar

[17]

R. Elie and I. Kharroubi, Probabilistic Representation and Approximation for couples systems of variational inequalities, Statistics and Probability Letters, 80 (2010), 1388-1396.  doi: 10.1016/j.spl.2010.05.003.  Google Scholar

[18]

M. Egami, A direct solution method for stochastic impulse control problems of one-dimensional diffusions, SIAM Journal on Control and Optimization, 47 (2008), 1191-1218.  doi: 10.1137/060669905.  Google Scholar

[19]

S. Hamadène and M. A. Morlais, Viscosity solutions of systems of pdes with interconnected obstacles and multi–modes switching problem, Applied Mathematics and Optimization, 67 (2013), 163–196. doi: 10.1007/s00245-012-9184-y.  Google Scholar

[20]

K. L. Helmes, R. H. Stockbridge and C. Zhu, A measure approach for continuous inventory models: Discounted cost criterion, SIAM Journal on Control and Optimization, 53 (2015), 2100-2140.  doi: 10.1137/140972640.  Google Scholar

[21]

K. Ishii, Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems, Funkcial. Ekvac., 36 (1993), 123-141.   Google Scholar

[22]

I. Kharroubi, J. Ma, H. Pham and J. Zhang, Backward SDEs with constrained jumps and quasi-variational inequalities, Ann. Probab., 38 (2010), 794-840.  doi: 10.1214/09-AOP496.  Google Scholar

[23]

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

[24]

S. M. Lenhart, Viscosity solutions associated with impulse control problems for piecewise deterministic processes, Internat. J. Math. Math. Sci., 12 (1989), 145-157.  doi: 10.1155/S0161171289000207.  Google Scholar

[25]

G. Mundaca and B. Oksendal, Optimal stochastic intervention control with application to the exchange rate, J. Math. Econom., 29 (1998), 225-243.  doi: 10.1016/S0304-4068(97)00013-X.  Google Scholar

[26]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition. Universitext. Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5.  Google Scholar

[27]

J. Palczewski and L. Stettner, Impulsive control of portfolios, Appl. Math. Optim., 56 (2007), 67-103.  doi: 10.1007/s00245-007-0880-y.  Google Scholar

[28]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion (Vol. 293)., Springer Science and Business Media, 2013. Google Scholar

[29]

R. C. Seydel, Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions, Stochastic Process. Appl., 119 (2009), 3719-3748.  doi: 10.1016/j.spa.2009.07.004.  Google Scholar

[30]

L. Stettner, Zero-sum Markov games with stopping and impulsive strategies, Appl. Math. Optim., 9 (1982), 1-24.  doi: 10.1007/BF01460115.  Google Scholar

[31]

S. J. Tang and J. M. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics Rep., 45 (1993), 145-176.  doi: 10.1080/17442509308833860.  Google Scholar

[32]

Y. Willassen, The stochastic rotation problem: A generalization of Faustmann's formula to stochastic forest growth, J. Econom. Dynam. Control, 22 (1998), 573-596.  doi: 10.1016/S0165-1889(97)00071-7.  Google Scholar

show all references

References:
[1]

L. H. Alvarez, Stochastic forest stand value and optimal timber harvesting, SIAM J. Control Optim., 42 (2004), 1972–1993 (electronic). doi: 10.1137/S0363012901393456.  Google Scholar

[2]

L. H. Alvarez, A class of solvable impulse control problems, Applied Mathematics and Optimization, 49 (2004), 265-295.  doi: 10.1007/s00245-004-0792-z.  Google Scholar

[3]

L. H. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732.  doi: 10.1137/060659375.  Google Scholar

[4]

P. Azimzadeh, Zero-sum stochastic differential game with impulses, precommitment and unrestricted cost functions, Applied Math. and Optim, 79 (2019), 483-514.  doi: 10.1007/s00245-017-9445-x.  Google Scholar

[5]

G. Barles and C. Imbert, Second order elliptic integro-differential Equations: Viscosity solutions's theory revisited., Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[6]

C. Belak, S. Christensen and F. T. Seifried, A general verification result for stochastic impulse control problems, SIAM J. Control Optim., 55 (2017), 627-649.  doi: 10.1137/16M1082822.  Google Scholar

[7]

A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, , Gauthier-Villars, Montrouge, 1984.  Google Scholar

[8]

B. Bouchard, A stochastic target formulation for optimal switching problems in finite horizon, Stochastics, 81 (2009), 171-197.  doi: 10.1080/17442500802327360.  Google Scholar

[9]

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.  Google Scholar

[10]

Y-S. A. Chen and X. Guo, Impulse control of multidimensional jump diffusions in finite time horison, SIAM J. Control Optim., 51 (2013), 2638-2663.  doi: 10.1137/110854205.  Google Scholar

[11]

M. Crandall, H. Ishii and P. L. Lions, Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

J. Dugundji, Topolgy, Boston: Allyn and Bacon, US, 1966.  Google Scholar

[13]

B. El Asri, Deterministic minimax impulse control in finite horizon: The viscosity solution approach., ESAIM: Control Optim. Calc. Var., 19 (2013), 63-77.  doi: 10.1051/cocv/2011200.  Google Scholar

[14]

B. El Asri, The value of a minimax problem involving impulse control, Journal of Dynamics and Games, 6 (20419), 1-17.  doi: 10.3934/jdg.2019001.  Google Scholar

[15]

B. El Asri and S. Mazid, Zero-sum stochastic differential game in finite horizon involving Impulse controls, Applied Mathematics and Optimization, 2018. doi: 10.1007/s00245-018-9529-2.  Google Scholar

[16]

B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, Journal of Mathematical Analysis and Applications, 474 (2019), 793-813.  doi: 10.1016/j.jmaa.2019.01.040.  Google Scholar

[17]

R. Elie and I. Kharroubi, Probabilistic Representation and Approximation for couples systems of variational inequalities, Statistics and Probability Letters, 80 (2010), 1388-1396.  doi: 10.1016/j.spl.2010.05.003.  Google Scholar

[18]

M. Egami, A direct solution method for stochastic impulse control problems of one-dimensional diffusions, SIAM Journal on Control and Optimization, 47 (2008), 1191-1218.  doi: 10.1137/060669905.  Google Scholar

[19]

S. Hamadène and M. A. Morlais, Viscosity solutions of systems of pdes with interconnected obstacles and multi–modes switching problem, Applied Mathematics and Optimization, 67 (2013), 163–196. doi: 10.1007/s00245-012-9184-y.  Google Scholar

[20]

K. L. Helmes, R. H. Stockbridge and C. Zhu, A measure approach for continuous inventory models: Discounted cost criterion, SIAM Journal on Control and Optimization, 53 (2015), 2100-2140.  doi: 10.1137/140972640.  Google Scholar

[21]

K. Ishii, Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems, Funkcial. Ekvac., 36 (1993), 123-141.   Google Scholar

[22]

I. Kharroubi, J. Ma, H. Pham and J. Zhang, Backward SDEs with constrained jumps and quasi-variational inequalities, Ann. Probab., 38 (2010), 794-840.  doi: 10.1214/09-AOP496.  Google Scholar

[23]

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

[24]

S. M. Lenhart, Viscosity solutions associated with impulse control problems for piecewise deterministic processes, Internat. J. Math. Math. Sci., 12 (1989), 145-157.  doi: 10.1155/S0161171289000207.  Google Scholar

[25]

G. Mundaca and B. Oksendal, Optimal stochastic intervention control with application to the exchange rate, J. Math. Econom., 29 (1998), 225-243.  doi: 10.1016/S0304-4068(97)00013-X.  Google Scholar

[26]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Second edition. Universitext. Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5.  Google Scholar

[27]

J. Palczewski and L. Stettner, Impulsive control of portfolios, Appl. Math. Optim., 56 (2007), 67-103.  doi: 10.1007/s00245-007-0880-y.  Google Scholar

[28]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion (Vol. 293)., Springer Science and Business Media, 2013. Google Scholar

[29]

R. C. Seydel, Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions, Stochastic Process. Appl., 119 (2009), 3719-3748.  doi: 10.1016/j.spa.2009.07.004.  Google Scholar

[30]

L. Stettner, Zero-sum Markov games with stopping and impulsive strategies, Appl. Math. Optim., 9 (1982), 1-24.  doi: 10.1007/BF01460115.  Google Scholar

[31]

S. J. Tang and J. M. Yong, Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach, Stochastics Rep., 45 (1993), 145-176.  doi: 10.1080/17442509308833860.  Google Scholar

[32]

Y. Willassen, The stochastic rotation problem: A generalization of Faustmann's formula to stochastic forest growth, J. Econom. Dynam. Control, 22 (1998), 573-596.  doi: 10.1016/S0165-1889(97)00071-7.  Google Scholar

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