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January  2019, 6(1): 1-17. doi: 10.3934/jdg.2019001

The value of a minimax problem involving impulse control

Brahim El Asri

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

Received  February 2018 Revised  December 2018 Published  January 2019

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.

Keywords: Impulse control, robust control, differential games, quasi-variational inequality, viscosity solution.
Mathematics Subject Classification: Primary: 49N25, 91G80, 49N70; Secondary: 35R45, 49L25.
Citation: Brahim El Asri. The value of a minimax problem involving impulse control. Journal of Dynamics & Games, 2019, 6 (1) : 1-17. doi: 10.3934/jdg.2019001
References:
[1]

V. I. Arnold, Ordinary Differential Equations, Springer, New York, 1992. Google Scholar

[2]

G. Barles, Deterministic impulse control problems, SIAM J. Control Optim., 23 (1985), 419-432.  doi: 10.1137/0323027.  Google Scholar

[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.  Google Scholar

[4]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Bordes, Paris, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. El Asri, Optimal multi-modes switching problem in infinite horizon, Stochastics and Dynamics, 10 (2010), 231-261.  doi: 10.1142/S0219493710002930.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, In arXiv preprint, 2018. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. H. Fleming, The convergence problem for differential games, Ⅱ., Ann. Math. Study, 52 (1964), 195-210.   Google Scholar

[20]

P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. M. Yong, Optimal switching and impulse controls for distributed parameter systems, Systems Sci Math Sci., 2 (1989), 137-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. M. Yong, Zero-sum differential games involving impulse controls, Appl.Math. Optim., 29 (1994), 243-261.  doi: 10.1007/BF01189477.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Ordinary Differential Equations, Springer, New York, 1992. Google Scholar

[2]

G. Barles, Deterministic impulse control problems, SIAM J. Control Optim., 23 (1985), 419-432.  doi: 10.1137/0323027.  Google Scholar

[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.  Google Scholar

[4]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Bordes, Paris, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. El Asri, Optimal multi-modes switching problem in infinite horizon, Stochastics and Dynamics, 10 (2010), 231-261.  doi: 10.1142/S0219493710002930.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. El Asri and S. Mazid, Stochastic differential switching game in infinite horizon, In arXiv preprint, 2018. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. H. Fleming, The convergence problem for differential games, Ⅱ., Ann. Math. Study, 52 (1964), 195-210.   Google Scholar

[20]

P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. M. Yong, Optimal switching and impulse controls for distributed parameter systems, Systems Sci Math Sci., 2 (1989), 137-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. M. Yong, Zero-sum differential games involving impulse controls, Appl.Math. Optim., 29 (1994), 243-261.  doi: 10.1007/BF01189477.  Google Scholar

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