République Algérienne Démocratique et Populaire
Ministère de l’Enseignement Supérieur et de la Recherche Scientifique
UNIVERSITÉ MOHAMED KHIDER, BISKRA
FACULTÉ des SCIENCES EXACTES et des SCIENCES de la NATURE et de la VIE
DÉPARTEMENT DE MATHÉMATIQUES.
THESE DE DOCTORAT EN SCIENCES
Option : Mathématiques
Par
Samira Boukaf
Titre
Sur un problem de contrôle optimal stochastique pour
certain aspect des équations differentielles stochastiques
de type mean-field et applications
Dr. Naceur Khelil, MCA, Université de Biskra, Président
Dr. Mokhtar Hafayed, MCA, Université de Biskra Rapporteur
Prof. Dahmen Achour, Prof. Université de M’sila Examinateur
Dr. Saadi Khalil, MCA, Université de Msila Examinateur
Dr. Boulakhras Gherbal , MCA, Université de Biskra, Examinateur
Dr. Abdelmouman Tiaiba, MCA. Université de M’sila Examinateur
2017
1
,Contents
I Introduction 10
1. Formulations of stochastic optimal control problems 10
1.1. Strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2. Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. Methods to solving optimal control problem 12
2.1. The Dynamic Programming Principle. . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2. The Pontryagin’s maximum principle . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Some classes of stochastic controls 18
II A study on optimal control problem with ελ −error bound for stochastic
systems with applications to linear quadratic problem 22
4. Introduction 22
5. Assumptions and Preliminaries 24
6. Stochastic maximum principle with ελ −error bound 25
7. Sufficient conditions for ε-optimality 32
8. Application: linear quadratic control problem 34
9. Concluding remarks and future research 35
III On Zhou’s maximum principle for near optimal control of mean-field
forward backward stochastic systems with jumps and its applications 37
10. Introduction 37
11. Formulation of the problem and preliminaries 39
12. Main results 44
12.1. Maximum principle of near-optimality for mean-field FBSDEJs . . . . . . . . . 44
12.2. Sufficient conditions for near-optimality of mean-field FBSDEJs . . . . . . . . . 58
13. Applications: Time-inconsistent mean-variance portfolio selection problem combined
with a recursive utility functional maximization 64
2
,IV Mean-field maximum principle for optimal control of forward-
backward stochastic systems with jumps and its application to mean-
variance portfolio problem 70
14. Introduction 70
15. Problem statement and preliminaries 73
16. Mean-field type necessary conditions for optimal control of FBSDEJs 75
17. Application: mean-variance portfolio selection problem mixed with a recursive utility
functional, time-inconsistent solution 85
V A McKean-Vlasov optimal mixed regular-singular control problem for
nonlinear stochastic systems with Poisson jump processes 92
18. Introduction 92
19. Assumptions and statement of the mixed control problem 95
20. Necessary conditions for optimal mixed continuous-singular control of McKean-Vlasov
FBSDEJs 101
21. Sufficient conditions for optimal mixed control of McKean-Vlasov FBSDEJs 107
22. Application: mean-variance portfolio selection problem with interventions control 113
VI Appendix 117
VII References 118
3
, I dedicate this work to my family.
4
Ministère de l’Enseignement Supérieur et de la Recherche Scientifique
UNIVERSITÉ MOHAMED KHIDER, BISKRA
FACULTÉ des SCIENCES EXACTES et des SCIENCES de la NATURE et de la VIE
DÉPARTEMENT DE MATHÉMATIQUES.
THESE DE DOCTORAT EN SCIENCES
Option : Mathématiques
Par
Samira Boukaf
Titre
Sur un problem de contrôle optimal stochastique pour
certain aspect des équations differentielles stochastiques
de type mean-field et applications
Dr. Naceur Khelil, MCA, Université de Biskra, Président
Dr. Mokhtar Hafayed, MCA, Université de Biskra Rapporteur
Prof. Dahmen Achour, Prof. Université de M’sila Examinateur
Dr. Saadi Khalil, MCA, Université de Msila Examinateur
Dr. Boulakhras Gherbal , MCA, Université de Biskra, Examinateur
Dr. Abdelmouman Tiaiba, MCA. Université de M’sila Examinateur
2017
1
,Contents
I Introduction 10
1. Formulations of stochastic optimal control problems 10
1.1. Strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2. Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. Methods to solving optimal control problem 12
2.1. The Dynamic Programming Principle. . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2. The Pontryagin’s maximum principle . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Some classes of stochastic controls 18
II A study on optimal control problem with ελ −error bound for stochastic
systems with applications to linear quadratic problem 22
4. Introduction 22
5. Assumptions and Preliminaries 24
6. Stochastic maximum principle with ελ −error bound 25
7. Sufficient conditions for ε-optimality 32
8. Application: linear quadratic control problem 34
9. Concluding remarks and future research 35
III On Zhou’s maximum principle for near optimal control of mean-field
forward backward stochastic systems with jumps and its applications 37
10. Introduction 37
11. Formulation of the problem and preliminaries 39
12. Main results 44
12.1. Maximum principle of near-optimality for mean-field FBSDEJs . . . . . . . . . 44
12.2. Sufficient conditions for near-optimality of mean-field FBSDEJs . . . . . . . . . 58
13. Applications: Time-inconsistent mean-variance portfolio selection problem combined
with a recursive utility functional maximization 64
2
,IV Mean-field maximum principle for optimal control of forward-
backward stochastic systems with jumps and its application to mean-
variance portfolio problem 70
14. Introduction 70
15. Problem statement and preliminaries 73
16. Mean-field type necessary conditions for optimal control of FBSDEJs 75
17. Application: mean-variance portfolio selection problem mixed with a recursive utility
functional, time-inconsistent solution 85
V A McKean-Vlasov optimal mixed regular-singular control problem for
nonlinear stochastic systems with Poisson jump processes 92
18. Introduction 92
19. Assumptions and statement of the mixed control problem 95
20. Necessary conditions for optimal mixed continuous-singular control of McKean-Vlasov
FBSDEJs 101
21. Sufficient conditions for optimal mixed control of McKean-Vlasov FBSDEJs 107
22. Application: mean-variance portfolio selection problem with interventions control 113
VI Appendix 117
VII References 118
3
, I dedicate this work to my family.
4
