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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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

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