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On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian
Stochastic impulse control Problem with state and time dependent cost functions
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 |
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.
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. 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. |
[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. 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. |
[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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