Abstract
In this paper, we consider a system of fully non linear second order parabolic partial differential equations with interconnected obstacles and boundary conditions on non smooth time-dependent domains. We prove existence and uniqueness of a continuous viscosity solution. This system is the HJB system of equations associated with a m-switching problem in finite horizon, when the state process is the solution of an obliquely reflected stochastic differential equation in non smooth time-dependent domain. Our approach is based on the study of related system of reflected generalized backward stochastic differential equations with oblique reflection. We show that this system has a unique solution which is the optimal payoff and provides the optimal strategy for the switching problem. Methods of the theory of generalized BSDEs and their connection with PDEs with boundary condition are then used to give a probabilistic representation for the solution of the PDE system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Not applicable
References
Boufoussi, B., Hamadène, S., Jakani, M.: Viscosity solutions of system of PDEs with interconnected obstacles and nonlinear Neumann boundary conditions. J. Math. Anal. Appl. 522, 126947 (2022)
Chassagneux, J.-F., Elie, R., Kharroubi, I.: A note on existence and uniqueness for solutions of multidimensional reflected BSDEs. Electron. Commun. Probab. 16, 120–128 (2011)
Costantini, C., Gobet, E., El Karoui, N.: Boundary sensitivities for diffusion processes in time dependent domains. Appl. Math. Optim. 54, 159–187 (2006)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide for viscosity solutions. Bull. Am. Math. Soc. 27(1), 1–67 (1992)
Djehiche, B., Hamadène, S., Popier, A.: A finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48(4), 2751–2770 (2009)
El Asri, B., Hamadène, S.: The finite horizon optimal multi-modes switching problem: the viscosity solution approach. Appl. Math. Optim. 60, 213–235 (2009)
Hamadène, S., Jeanblanc, M.: On the starting and stopping problem: application in reversible investments. Math. Oper. Res. 32(1), 182–192 (2007)
Hamadène, S., Mnif, M., Neffati, S.: Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition. Asymptotic Anal. 113(3), 123–136 (2019)
Hamadène, S., Morlais, M.A.: Viscosity solutions of systems of PDEs with interconnected obstacles and multi-modes switching problem. Appl. Math. Optim. 67(2), 163–196 (2013)
Hamadène, S., Zhang, J.: Switching problem and related system of reflected backward stochastic differential equations. Stoch. Process. Appl. 120, 403–426 (2010)
Hamadène, S., Zhao, X.: Systems of integro-PDEs with interconnected obstacles and multi-modes switching problem driven by Lévy process. NoDEA Nonlinear Differ. Equ. Appl. 22(6), 1607–1660 (2015)
Hu, Y., Tang, S.: Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147(1–2), 89–121 (2010)
Jakani, M.: Systems of PDEs with interconnected bilateral obstacles and nonlinear Neumann boundary conditions . (hal-03603191) (2022)
Li, J., Tang, S.: Optimal stochastic control with recursive cost functional of stochastic differential systems reflected in a domain, arXiv:1202.1412v3
Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537 (1984)
Lundström, N.L.P., Olofsson, M.: Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions. Appl. Math. Optim. 86, 27 (2022)
Lundström, N.L.P., Olofsson, M., Önskog, T.: Management strategies for run-of-river hydropower plants: an optimal switching approach. Optim. Eng. 23, 1707–1731 (2021)
Lundström, N.L.P., Önskog, T.: Stochastic and partial differential equations on nonsmooth time-dependent domains. Stoch. Process. Appl. 129(4), 1097–1131 (2019)
Pardoux, E., Rascanu, A.: Continuity of the Feynman-Kac formula for a generalized parabolic equation. Stochastics 89(5), 726–752 (2017)
Pardoux, E., Zhang, S.: Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields 110(4), 535–558 (1998)
Ren, Y., Xia, N.: Generalized reflected BSDE and obstacle problem for PDE with nonlinear Neumann boundary condition. Stoch. Anal. Appl. 24, 1013–1033 (2006)
Acknowledgements
The author thanks the referees for carefully reading the paper and for providing valuable suggestions and comments.
Funding
The author was supported by [Campus France] (Grant ID [PHC Toubkal/18/59]).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no conflict of interest to declare.
Ethical Approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jakani, M. System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains. Appl Math Optim 88, 31 (2023). https://doi.org/10.1007/s00245-023-10012-6
Accepted:
Published:
DOI: https://doi.org/10.1007/s00245-023-10012-6
Keywords
- Viscosity solution
- Fully nonlinear partial differential equations
- Obliquely reflected diffusion
- Non-smooth time-dependent domain
- Generalized Backward stochastic differential systems
- Optimal switching