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

The value of a minimax problem involving impulse control

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.

Citation: Brahim El Asri. The value of a minimax problem involving impulse control. Journal of Dynamics and Games, 2019, 6 (1) : 1-17. doi: 10.3934/jdg.2019001
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. BarronL. 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. BernhardN. 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. BertolaW. 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. CrandallH. 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 FarouqG. 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. BarronL. 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. BernhardN. 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. BertolaW. 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. CrandallH. 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 FarouqG. 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.

[1]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101

[2]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165

[3]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523

[4]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457

[5]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583

[6]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383

[7]

Alain Bensoussan, John Liu, Jiguang Yuan. Singular control and impulse control: A common approach. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 27-57. doi: 10.3934/dcdsb.2010.13.27

[8]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423

[9]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial and Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363

[10]

Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331

[11]

Weihua Ruan. Markovian strategies for piecewise deterministic differential games with continuous and impulse controls. Journal of Dynamics and Games, 2019, 6 (4) : 337-366. doi: 10.3934/jdg.2019022

[12]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989

[13]

Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131

[14]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105

[15]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control and Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365

[16]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[17]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure and Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583

[18]

T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437

[19]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1115-1132. doi: 10.3934/jimo.2021011

[20]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

 Impact Factor: 

Metrics

  • PDF downloads (264)
  • HTML views (345)
  • Cited by (0)

Other articles
by authors

[Back to Top]