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The value of a minimax problem involving impulse control
Université Ibn Zohr, Equipe. Aide à la decision, ENSA, B.P. 1136, Agadir, Maroc |
We consider the minimax impulse control problem in finite horizon, when the cost functions are positive and not bounded from below with a strictly positive constant. We show existence of value function of the problem. Moreover, the value function is characterized as the unique viscosity solution of Hamilton-Jacobi-Bellman-Isaacs equation. This problem is in relation with an application in mathematical finance.
References:
| [1] |
V. I. Arnold, Ordinary Differential Equations, Springer, New York, 1992. |
| [2] |
G. Barles,
Deterministic impulse control problems, SIAM J. Control Optim., 23 (1985), 419-432.
doi: 10.1137/0323027. |
| [3] |
E. N. Barron, L. C. Evans and R. Jensen,
Viscosity solutions of Isaaes' equations and differential games with Lipschitz controls, J Differential Equations, 53 (1984), 213-233.
doi: 10.1016/0022-0396(84)90040-8. |
| [4] |
A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Bordes, Paris, 1984. |
| [5] |
P. Bernhard,
A robust control approach to option pricing including transaction costs, Annals of the ISDG., 7 (2005), 391-416.
doi: 10.1007/0-8176-4429-6_22. |
| [6] |
P. Bernhard, N. El Farouq and S. Thiery,
An impulsive differential game arising in finance with interesting singularities, Annals of the ISDG., 8 (2006), 335-363.
doi: 10.1007/0-8176-4501-2_18. |
| [7] |
G. Bertola, W. Runggaldier and K. Yasuda,
On classical and restricted impulse stochastic control for the exchange rate, Appl Math Optim., 74 (2016), 423-454.
doi: 10.1007/s00245-015-9320-6. |
| [8] |
I. Capuzzo-Dolcetta and L. C. Evans,
Optimal switching for ordinary differential equations, SIAM J. Control Optim., 22 (1984), 143-161.
doi: 10.1137/0322011. |
| [9] |
M. Crandall, H. Ishii and P. L. Lions,
User s 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. |
| [10] |
S. Dharmatti and A. J. Shaiju,
Infinite dimensional differential games with hybrid controls, Proc. Indian Acad. Sci. Math., 117 (2007), 233-257.
doi: 10.1007/s12044-007-0019-8. |
| [11] |
S. Dharmatti and M. Ramaswamy,
Zero-sum differential games involving hybrid controls, J. Optim. Theory Appl., 128 (2006), 75-102.
doi: 10.1007/s10957-005-7558-x. |
| [12] |
B. El Asri,
Optimal multi-modes switching problem in infinite horizon, Stochastics and Dynamics, 10 (2010), 231-261.
doi: 10.1142/S0219493710002930. |
| [13] |
B. El Asri,
Deterministic minimax impulse control in finite horizon: The viscosity solution approach, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 63-77.
doi: 10.1051/cocv/2011200. |
| [14] |
B. El Asri,
Stochastic optimal multi-modes switching with a viscosity solution approach, Stochastic Processes and their Applications, 123 (2013), 579-602.
doi: 10.1016/j.spa.2012.09.007. |
| [15] |
B. EL Asri and S. Mazid, Zero-sum stochastic differential game in finite horizon involving impulse controls, Appl Math Optim., (2018), 1-33.
doi: 10.1007/s00245-018-9529-2. |
| [16] |
B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, In arXiv preprint, 2018. |
| [17] |
N. El Farouq, G. Barles and P. Bernhard,
Deterministic minimax impulse control, Appl Math Optim., 61 (2010), 353-378.
doi: 10.1007/s00245-009-9090-0. |
| [18] |
L. C. Evans and P. E. Souganidis,
Differential games and representation formulas for the solution of Hamilton-Jacobi-Isaacs equations, Indiana Univ. J. Math., 33 (1984), 773-797.
doi: 10.1512/iumj.1984.33.33040. |
| [19] |
W. H. Fleming,
The convergence problem for differential games, Ⅱ., Ann. Math. Study, 52 (1964), 195-210.
|
| [20] |
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. |
| [21] |
P. L. Lions and P. E. Souganidis,
Differential games, optimal control and directional derivatives of viscosity solutions of Bellman s and Isaacs equations, SIAM J. Control Optim., 23 (1985), 566-583.
doi: 10.1137/0323036. |
| [22] |
A. J. Shaiju and S. Dharmatti,
Differential games with continuous, switching and impulse controls, Nonlinear Anal., 63 (2005), 23-41.
doi: 10.1016/j.na.2005.04.002. |
| [23] |
P. E. Souganidis,
Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games, Nonlinear Anal. Theory Methods Appl., 9 (1985), 217-257.
doi: 10.1016/0362-546X(85)90062-8. |
| [24] |
J. M. Yong,
Systems governed by ordinary differential equations with continuous, switching and impulse controls, Appl Math Opti., 20 (1989), 223-235.
doi: 10.1007/BF01447655. |
| [25] |
J. M. Yong,
Optimal switching and impulse controls for distributed parameter systems, Systems Sci Math Sci., 2 (1989), 137-160.
|
| [26] |
J. M. Yong,
Differential games with switching strategies, J Math Anal Appl., 145 (1990), 455-469.
doi: 10.1016/0022-247X(90)90413-A. |
| [27] |
J. M. Yong,
A zero-sum differential game in a finite duration with switching strategies, SIAM J Control Optim., 28 (1990), 1234-1250.
doi: 10.1137/0328066. |
| [28] |
J. M. Yong,
Zero-sum differential games involving impulse controls, Appl.Math. Optim., 29 (1994), 243-261.
doi: 10.1007/BF01189477. |
show all references
References:
| [1] |
V. I. Arnold, Ordinary Differential Equations, Springer, New York, 1992. |
| [2] |
G. Barles,
Deterministic impulse control problems, SIAM J. Control Optim., 23 (1985), 419-432.
doi: 10.1137/0323027. |
| [3] |
E. N. Barron, L. C. Evans and R. Jensen,
Viscosity solutions of Isaaes' equations and differential games with Lipschitz controls, J Differential Equations, 53 (1984), 213-233.
doi: 10.1016/0022-0396(84)90040-8. |
| [4] |
A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Bordes, Paris, 1984. |
| [5] |
P. Bernhard,
A robust control approach to option pricing including transaction costs, Annals of the ISDG., 7 (2005), 391-416.
doi: 10.1007/0-8176-4429-6_22. |
| [6] |
P. Bernhard, N. El Farouq and S. Thiery,
An impulsive differential game arising in finance with interesting singularities, Annals of the ISDG., 8 (2006), 335-363.
doi: 10.1007/0-8176-4501-2_18. |
| [7] |
G. Bertola, W. Runggaldier and K. Yasuda,
On classical and restricted impulse stochastic control for the exchange rate, Appl Math Optim., 74 (2016), 423-454.
doi: 10.1007/s00245-015-9320-6. |
| [8] |
I. Capuzzo-Dolcetta and L. C. Evans,
Optimal switching for ordinary differential equations, SIAM J. Control Optim., 22 (1984), 143-161.
doi: 10.1137/0322011. |
| [9] |
M. Crandall, H. Ishii and P. L. Lions,
User s 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. |
| [10] |
S. Dharmatti and A. J. Shaiju,
Infinite dimensional differential games with hybrid controls, Proc. Indian Acad. Sci. Math., 117 (2007), 233-257.
doi: 10.1007/s12044-007-0019-8. |
| [11] |
S. Dharmatti and M. Ramaswamy,
Zero-sum differential games involving hybrid controls, J. Optim. Theory Appl., 128 (2006), 75-102.
doi: 10.1007/s10957-005-7558-x. |
| [12] |
B. El Asri,
Optimal multi-modes switching problem in infinite horizon, Stochastics and Dynamics, 10 (2010), 231-261.
doi: 10.1142/S0219493710002930. |
| [13] |
B. El Asri,
Deterministic minimax impulse control in finite horizon: The viscosity solution approach, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 63-77.
doi: 10.1051/cocv/2011200. |
| [14] |
B. El Asri,
Stochastic optimal multi-modes switching with a viscosity solution approach, Stochastic Processes and their Applications, 123 (2013), 579-602.
doi: 10.1016/j.spa.2012.09.007. |
| [15] |
B. EL Asri and S. Mazid, Zero-sum stochastic differential game in finite horizon involving impulse controls, Appl Math Optim., (2018), 1-33.
doi: 10.1007/s00245-018-9529-2. |
| [16] |
B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, In arXiv preprint, 2018. |
| [17] |
N. El Farouq, G. Barles and P. Bernhard,
Deterministic minimax impulse control, Appl Math Optim., 61 (2010), 353-378.
doi: 10.1007/s00245-009-9090-0. |
| [18] |
L. C. Evans and P. E. Souganidis,
Differential games and representation formulas for the solution of Hamilton-Jacobi-Isaacs equations, Indiana Univ. J. Math., 33 (1984), 773-797.
doi: 10.1512/iumj.1984.33.33040. |
| [19] |
W. H. Fleming,
The convergence problem for differential games, Ⅱ., Ann. Math. Study, 52 (1964), 195-210.
|
| [20] |
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. |
| [21] |
P. L. Lions and P. E. Souganidis,
Differential games, optimal control and directional derivatives of viscosity solutions of Bellman s and Isaacs equations, SIAM J. Control Optim., 23 (1985), 566-583.
doi: 10.1137/0323036. |
| [22] |
A. J. Shaiju and S. Dharmatti,
Differential games with continuous, switching and impulse controls, Nonlinear Anal., 63 (2005), 23-41.
doi: 10.1016/j.na.2005.04.002. |
| [23] |
P. E. Souganidis,
Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games, Nonlinear Anal. Theory Methods Appl., 9 (1985), 217-257.
doi: 10.1016/0362-546X(85)90062-8. |
| [24] |
J. M. Yong,
Systems governed by ordinary differential equations with continuous, switching and impulse controls, Appl Math Opti., 20 (1989), 223-235.
doi: 10.1007/BF01447655. |
| [25] |
J. M. Yong,
Optimal switching and impulse controls for distributed parameter systems, Systems Sci Math Sci., 2 (1989), 137-160.
|
| [26] |
J. M. Yong,
Differential games with switching strategies, J Math Anal Appl., 145 (1990), 455-469.
doi: 10.1016/0022-247X(90)90413-A. |
| [27] |
J. M. Yong,
A zero-sum differential game in a finite duration with switching strategies, SIAM J Control Optim., 28 (1990), 1234-1250.
doi: 10.1137/0328066. |
| [28] |
J. M. Yong,
Zero-sum differential games involving impulse controls, Appl.Math. Optim., 29 (1994), 243-261.
doi: 10.1007/BF01189477. |
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