June  2020, 10(2): 425-441. doi: 10.3934/mcrf.2020004

Singular control of SPDEs with space-mean dynamics

1. 

Department of Mathematics, Linnaeus University (LNU), Sweden

2. 

Department of Mathematics, University of Oslo, Norway

* Corresponding author: Bernt Øksendal

Received  May 2019 Revised  June 2019 Published  June 2020 Early access  November 2019

Fund Project: This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for these control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

Citation: Nacira Agram, Astrid Hilbert, Bernt Øksendal. Singular control of SPDEs with space-mean dynamics. Mathematical Control and Related Fields, 2020, 10 (2) : 425-441. doi: 10.3934/mcrf.2020004
References:
[1]

N. Agram, A. Hilbert and B. Øksendal, SPDEs with space-mean dynamics, preprint, arXiv: math/1807.07303.

[2]

A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), 169-222.  doi: 10.1080/17442508308833253.

[3]

A. Bensoussan, Stochastic maximum principle for systems with partial information and application to the separation principle, in Applied Stochastic Analysis, Stochastics Monogr., 5, Gordon and Breach, New York, 1991,157–172.

[4] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.
[5]

C. Donati-Martin and É. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 152, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[7]

L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimension, Probability and its Applications, Springer Heidelberg Dordrecht London New York, 2011. doi: 10.1017/CBO9781107295513.

[8]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Universitext, Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[9]

Y. HuJ. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.

[10]

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

[11]

M. Hairer, An introduction to stochastic PDEs, preprint, arXiv: math/0907.4178.

[12]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.

[13]

B. Øksendal, Optimal control of stochastic partial differential equations, Stoch. Anal. Appl., 23 (2005), 165-179.  doi: 10.1081/SAP-200044467.

[14]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Universitext, Springer, Cham, 2019. doi: 10.1007/978-3-030-02781-0.

[15]

B. ØksendalF. Proske and T. Zhang, Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields, Stochastics, 77 (2005), 381-399.  doi: 10.1080/17442500500213797.

[16]

B. ØksendalA. Sulem and T. Zhang, Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection, Math. Oper. Res., 39 (2014), 464-486.  doi: 10.1287/moor.2013.0602.

[17]

E. Pardouxt, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.

[18]

E. Pardoux, Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, in École d'Été Probabilites de Saint-Flour XIX, Lecture Notes in Math., 1464, Springer, Berlin, 1991, 67–163.

[19]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.

[20]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.

[21] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.

show all references

References:
[1]

N. Agram, A. Hilbert and B. Øksendal, SPDEs with space-mean dynamics, preprint, arXiv: math/1807.07303.

[2]

A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), 169-222.  doi: 10.1080/17442508308833253.

[3]

A. Bensoussan, Stochastic maximum principle for systems with partial information and application to the separation principle, in Applied Stochastic Analysis, Stochastics Monogr., 5, Gordon and Breach, New York, 1991,157–172.

[4] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.
[5]

C. Donati-Martin and É. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields, 95 (1993), 1-24.  doi: 10.1007/BF01197335.

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 152, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[7]

L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimension, Probability and its Applications, Springer Heidelberg Dordrecht London New York, 2011. doi: 10.1017/CBO9781107295513.

[8]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Universitext, Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[9]

Y. HuJ. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.

[10]

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

[11]

M. Hairer, An introduction to stochastic PDEs, preprint, arXiv: math/0907.4178.

[12]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.

[13]

B. Øksendal, Optimal control of stochastic partial differential equations, Stoch. Anal. Appl., 23 (2005), 165-179.  doi: 10.1081/SAP-200044467.

[14]

B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Universitext, Springer, Cham, 2019. doi: 10.1007/978-3-030-02781-0.

[15]

B. ØksendalF. Proske and T. Zhang, Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields, Stochastics, 77 (2005), 381-399.  doi: 10.1080/17442500500213797.

[16]

B. ØksendalA. Sulem and T. Zhang, Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection, Math. Oper. Res., 39 (2014), 464-486.  doi: 10.1287/moor.2013.0602.

[17]

E. Pardouxt, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.

[18]

E. Pardoux, Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, in École d'Été Probabilites de Saint-Flour XIX, Lecture Notes in Math., 1464, Springer, Berlin, 1991, 67–163.

[19]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007/978-3-540-70781-3.

[20]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.

[21] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
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