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September  2020, 9(3): 817-843. doi: 10.3934/eect.2020035

On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application

Adel Chala , and  Dahbia Hafayed

Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145, Biskra 07000. Algeria

* Corresponding author: Adel Chala

Received  October 2018 Revised  October 2019 Published  September 2020 Early access  March 2020

Fund Project: The first author is supported by PRFU project N: C00L03UN070120180002

In this paper, we are concerned with an optimal control problem where the system is driven by fully coupled forward-backward stochastic differential equation of mean-field type with risk-sensitive performance functional. We study the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of the initial stochastic control problem in this type of risk-sensitive performance problem, where an admissible set of controls are convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. Finally, we illustrate our main result of this paper by giving two examples of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function, the second example will be a mean-variance portfolio with a recursive utility functional optimization problem involving optimal control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback.

Keywords: Fully coupled forward-backward stochastic differential equation of mean-field type, risk-sensitive, stochastic maximum principle, optimal control, variational principle, logarithmic transformation, mean-variance.
Mathematics Subject Classification: Primary: 93E20, 60H30; Secondary: 60H10, 91B28.
Citation: Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
References:
[1]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[2]

F. Armerin, Aspects of cash flow valuation, Ph.D thesis, Kungliga Tekniska Hogskolan (Sweden), 2004,116 pp.

[3]

R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.

[4]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[5]

R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[6]

R. Carmona and F. Delarue, Mean-field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15pp. doi: 10.1214/ECP.v18-2446.

[7]

A. Chala, Pontryagin's risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.), 48 (2017), 399-411.  doi: 10.1007/s00574-017-0031-2.

[8]

A. Chala, Sufficient optimality condition for a risk-sensitive control problem for backward stochastic differential equations and an application, J. Numer. Math. Stoch., 09 (2017), 48-60. 

[9]

A. Chala, D. Hafayed and R. Khallout, The use of Girsanov's theorem to describe the risk-sensitive problem and application to optimal control, in Stochastic Differential Equation-Basics and Applications, Nova Science Publishers, Inc., 2018,111–142.

[10]

B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control, 60 (2015), 2640-2649.  doi: 10.1109/TAC.2015.2406973.

[11]

N. El-Karoui and S. Hamadène, BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Process. Appl., 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0.

[12]

D. Hafayed and A. Chala, An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications, Random Operators and Stochastic Equation, published online, (2020). doi: 10.1515/rose-2020-2024.

[13]

Y. Hu, B. Øksendal and A. Sulem, Singular mean-field control games with applications to optimal harvesting and investment problems, preprint, arXiv: 1406.1863, (2014).

[14]

J. M. Lasry and P. L. Lions, Mean-field games, Jpn. J. Math., 02 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[15]

J. Li, Stochastic maximum principle in the mean-field controls, Automatica J. IFAC, 48 (2012), 366-373.  doi: 10.1016/j.automatica.2011.11.006.

[16]

A. E. B. Lim and X. Y. Zhou, A new risk-sensitive maximum principle, IEEE Trans. Automat. Control, 50 (2005), 958-966.  doi: 10.1109/TAC.2005.851441.

[17]

T. Meyer-Brandis, B. Øksendal and X. Y. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666.  doi: 10.1080/17442508.2011.651619.

[18]

H. Min, Y. Peng and Y. Qin, Fully coupled mean-field forward-backward stochastic differential equations and stochastic maximum principle, Abstr. Appl. Anal., 2014 (2014), Art. ID 839467, 15 pp. doi: 10.1155/2014/839467.

[19]

J. Shi and Z. Wu, A risk-sensitive stochastic maximum principle for optimal control of jump diffusions and its applications, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 419-433.  doi: 10.1016/S0252-9602(11)60242-7.

[20]

J. Shi and Z. Wu, Maximum principle for risk-sensitive stochastic optimal control problem and applications to finance, Stoch. Anal. Appl., 30 (2012), 997-1018.  doi: 10.1080/07362994.2012.727138.

[21]

A. S. Sznitman, Topics in propagation of chaos, in In Ecole d'Été de Probabilités de Saint-Flour XIX–1989, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169.

[22]

H. Tembine, Risk-sensitive mean-field-type games with Lp-norm drifts, Automatica J. IFAC, 59 (2015), 224-237.  doi: 10.1016/j.automatica.2015.06.036.

[23]

J. Yong, A stochastic linear quadratic optimal control problem with generalized expectation, Stoch. Anal. Appl., 26 (2008), 1136-1160.  doi: 10.1080/07362990802286533.

[24]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[25]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.

show all references

References:
[1]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[2]

F. Armerin, Aspects of cash flow valuation, Ph.D thesis, Kungliga Tekniska Hogskolan (Sweden), 2004,116 pp.

[3]

R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.

[4]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[5]

R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[6]

R. Carmona and F. Delarue, Mean-field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15pp. doi: 10.1214/ECP.v18-2446.

[7]

A. Chala, Pontryagin's risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.), 48 (2017), 399-411.  doi: 10.1007/s00574-017-0031-2.

[8]

A. Chala, Sufficient optimality condition for a risk-sensitive control problem for backward stochastic differential equations and an application, J. Numer. Math. Stoch., 09 (2017), 48-60. 

[9]

A. Chala, D. Hafayed and R. Khallout, The use of Girsanov's theorem to describe the risk-sensitive problem and application to optimal control, in Stochastic Differential Equation-Basics and Applications, Nova Science Publishers, Inc., 2018,111–142.

[10]

B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control, 60 (2015), 2640-2649.  doi: 10.1109/TAC.2015.2406973.

[11]

N. El-Karoui and S. Hamadène, BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Process. Appl., 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0.

[12]

D. Hafayed and A. Chala, An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications, Random Operators and Stochastic Equation, published online, (2020). doi: 10.1515/rose-2020-2024.

[13]

Y. Hu, B. Øksendal and A. Sulem, Singular mean-field control games with applications to optimal harvesting and investment problems, preprint, arXiv: 1406.1863, (2014).

[14]

J. M. Lasry and P. L. Lions, Mean-field games, Jpn. J. Math., 02 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[15]

J. Li, Stochastic maximum principle in the mean-field controls, Automatica J. IFAC, 48 (2012), 366-373.  doi: 10.1016/j.automatica.2011.11.006.

[16]

A. E. B. Lim and X. Y. Zhou, A new risk-sensitive maximum principle, IEEE Trans. Automat. Control, 50 (2005), 958-966.  doi: 10.1109/TAC.2005.851441.

[17]

T. Meyer-Brandis, B. Øksendal and X. Y. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666.  doi: 10.1080/17442508.2011.651619.

[18]

H. Min, Y. Peng and Y. Qin, Fully coupled mean-field forward-backward stochastic differential equations and stochastic maximum principle, Abstr. Appl. Anal., 2014 (2014), Art. ID 839467, 15 pp. doi: 10.1155/2014/839467.

[19]

J. Shi and Z. Wu, A risk-sensitive stochastic maximum principle for optimal control of jump diffusions and its applications, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 419-433.  doi: 10.1016/S0252-9602(11)60242-7.

[20]

J. Shi and Z. Wu, Maximum principle for risk-sensitive stochastic optimal control problem and applications to finance, Stoch. Anal. Appl., 30 (2012), 997-1018.  doi: 10.1080/07362994.2012.727138.

[21]

A. S. Sznitman, Topics in propagation of chaos, in In Ecole d'Été de Probabilités de Saint-Flour XIX–1989, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169.

[22]

H. Tembine, Risk-sensitive mean-field-type games with Lp-norm drifts, Automatica J. IFAC, 59 (2015), 224-237.  doi: 10.1016/j.automatica.2015.06.036.

[23]

J. Yong, A stochastic linear quadratic optimal control problem with generalized expectation, Stoch. Anal. Appl., 26 (2008), 1136-1160.  doi: 10.1080/07362990802286533.

[24]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[25]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.

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