American Institute of Mathematical Sciences

Advanced
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

[1]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[2]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[3]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020
[4]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[5]

Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011
[6]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072
[7]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[8]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[9]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[10]

Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021069
[11]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026
[12]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[13]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[14]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[15]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[16]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215
[17]

Yi Gao, Rui Li, Yingjing Shi, Li Xiao. Design of path planning and tracking control of quadrotor. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021063
[18]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[19]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[20]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

 Impact Factor: 

Tools

Metrics

  • PDF downloads (131)
  • HTML views (344)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences