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December  2021, 11(4): 829-855. doi: 10.3934/mcrf.2020048

Stochastic maximum principle for problems with delay with dependence on the past through general measures

Giuseppina Guatteri 1, and  Federica Masiero 2,,

1. 

Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italia
2. 

Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via Cozzi 55, 20125 Milano, Italia

* Corresponding author: Federica Masiero

Received  February 2020 Revised  September 2020 Published  December 2021 Early access  November 2020

We prove a stochastic maximum principle for a control problem where the state equation is delayed both in the state and in the control, and both the running and the final cost functionals may depend on the past trajectories. The adjoint equation turns out to be a new form of linear anticipated backward stochastic differential equations (ABSDEs in the following), and we prove a direct formula to solve these equations.

Keywords: Stochastic maximum principle, delay, anticipated backward stochastic differential equations.
Mathematics Subject Classification: 60H10, 93E20.
Citation: Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048
References:
[1]

E. Bandini, A. Cosso, M. Fuhrman and H. Pham, Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach, Ann. Appl. Probab., 28 (2018), 1634-1678.  doi: 10.1214/17-AAP1340.

[2]

B. Bruder and H. Pham, Impulse control problem on finite horizon with execution delay, Stochastic Process. Appl., 119 (2009), 1436-1469.  doi: 10.1016/j.spa.2008.07.007.

[3]

R. Buckdahn, H.-J. Engelbert and A. Răşcanu, On weak solutions of backward stochastic differential equations, Teor. Veroyatn. Primen., 49 (2004), 70-108.  doi: 10.4213/tvp237.

[4]

L. Chen, Z. Wu and Z. Yu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080.  doi: 10.1016/j.automatica.2010.03.005.

[5]

L. Chen and Z. Wu, Delayed stochastic linear-quadratic control problem and related applications, Journal of Applied Mathematics, 2012 (2012), 835319, 22 pp. doi: 10.1155/2012/835319.

[6]

Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis, An Introduction, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/gsm/066.

[7]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[8]

G. Fabbri and S. Federico, On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term, Mathematical Economics Letters, 2 (2014), 33-44.  doi: 10.1515/mel-2014-0011.

[9]

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.  doi: 10.1214/aop/1029867132.

[10]

F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, Stochastic partial differential equations and applications—VII, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 245 (2006), 133–148.

[11]

F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, J. Optim. Theory Appl., 142 (2009), 291-321.  doi: 10.1007/s10957-009-9524-5.

[12]

F. Gozzi and F. Masiero, Stochastic optimal control with delay in the control II:Verification theorem and optimal feedbacks, SIAM J. Control Optim., 55 (2017), 3013-3038.  doi: 10.1137/16M1073637.

[13]

L. Grosset and B. Viscolani, Advertising for a new product introduction: A stochastic approach, Top, 12 (2004), 149-167.  doi: 10.1007/BF02578929.

[14]

G. Guatteri, F. Masiero and C. Orrieri, Stochastic maximum principle for SPDEs with delay, Stochastic Process. Appl., 127 (2017), 2396-2427.  doi: 10.1016/j.spa.2016.11.007.

[15]

R. F. Hartl, Optimal dynamic advertising policies for hereditary processes, J. Optim. Theory Appl., 43 (1984), 51-72.  doi: 10.1007/BF00934746.

[16]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.  doi: 10.1080/17442509008833671.

[17]

Y. Hu and S. Peng, Maximum principle for optimal control of stochastic system of functional type, Stochastic Anal. Appl., 14 (1996), 283-301.  doi: 10.1080/07362999608809440.

[18]

S.-E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications. Stochastic analysis and related topics VI, Progr. Probab., Birkhäuser Boston, Boston, MA, 42 (1998), 1–77.

[19]

B. Øksendal, A. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. in Appl. Probab., 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.

[20]

C. Orrieri, E. Rocca and L. Scarpa, Optimal control of stochastic phase-field models related to tumor growth, ESAIM Control Optim. Calc. Var., forthcoming. doi: 10.1051/cocv/2020022.

[21]

E. Pardoux and A. Răşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, Springer, Cham, 2014. doi: 10.1007/978-3-319-05714-9.

[22]

S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.  doi: 10.1214/08-AOP423.

[23]

Z. Yang and R. J. Elliott, Some properties of generalized anticipated backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 10. doi: 10.1214/ECP.v18-2415.

[24]

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

show all references

References:
[1]

E. Bandini, A. Cosso, M. Fuhrman and H. Pham, Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach, Ann. Appl. Probab., 28 (2018), 1634-1678.  doi: 10.1214/17-AAP1340.

[2]

B. Bruder and H. Pham, Impulse control problem on finite horizon with execution delay, Stochastic Process. Appl., 119 (2009), 1436-1469.  doi: 10.1016/j.spa.2008.07.007.

[3]

R. Buckdahn, H.-J. Engelbert and A. Răşcanu, On weak solutions of backward stochastic differential equations, Teor. Veroyatn. Primen., 49 (2004), 70-108.  doi: 10.4213/tvp237.

[4]

L. Chen, Z. Wu and Z. Yu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080.  doi: 10.1016/j.automatica.2010.03.005.

[5]

L. Chen and Z. Wu, Delayed stochastic linear-quadratic control problem and related applications, Journal of Applied Mathematics, 2012 (2012), 835319, 22 pp. doi: 10.1155/2012/835319.

[6]

Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis, An Introduction, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/gsm/066.

[7]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[8]

G. Fabbri and S. Federico, On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term, Mathematical Economics Letters, 2 (2014), 33-44.  doi: 10.1515/mel-2014-0011.

[9]

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.  doi: 10.1214/aop/1029867132.

[10]

F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, Stochastic partial differential equations and applications—VII, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 245 (2006), 133–148.

[11]

F. Gozzi, C. Marinelli and S. Savin, On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects, J. Optim. Theory Appl., 142 (2009), 291-321.  doi: 10.1007/s10957-009-9524-5.

[12]

F. Gozzi and F. Masiero, Stochastic optimal control with delay in the control II:Verification theorem and optimal feedbacks, SIAM J. Control Optim., 55 (2017), 3013-3038.  doi: 10.1137/16M1073637.

[13]

L. Grosset and B. Viscolani, Advertising for a new product introduction: A stochastic approach, Top, 12 (2004), 149-167.  doi: 10.1007/BF02578929.

[14]

G. Guatteri, F. Masiero and C. Orrieri, Stochastic maximum principle for SPDEs with delay, Stochastic Process. Appl., 127 (2017), 2396-2427.  doi: 10.1016/j.spa.2016.11.007.

[15]

R. F. Hartl, Optimal dynamic advertising policies for hereditary processes, J. Optim. Theory Appl., 43 (1984), 51-72.  doi: 10.1007/BF00934746.

[16]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.  doi: 10.1080/17442509008833671.

[17]

Y. Hu and S. Peng, Maximum principle for optimal control of stochastic system of functional type, Stochastic Anal. Appl., 14 (1996), 283-301.  doi: 10.1080/07362999608809440.

[18]

S.-E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications. Stochastic analysis and related topics VI, Progr. Probab., Birkhäuser Boston, Boston, MA, 42 (1998), 1–77.

[19]

B. Øksendal, A. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. in Appl. Probab., 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.

[20]

C. Orrieri, E. Rocca and L. Scarpa, Optimal control of stochastic phase-field models related to tumor growth, ESAIM Control Optim. Calc. Var., forthcoming. doi: 10.1051/cocv/2020022.

[21]

E. Pardoux and A. Răşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, Springer, Cham, 2014. doi: 10.1007/978-3-319-05714-9.

[22]

S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.  doi: 10.1214/08-AOP423.

[23]

Z. Yang and R. J. Elliott, Some properties of generalized anticipated backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 10. doi: 10.1214/ECP.v18-2415.

[24]

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

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