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System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics & Optimization Springer Journals

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Jakani, Manal
Applied Mathematics & Optimization , Volume 88 (2) – Oct 1, 2023
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References (22)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. 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.
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-023-10012-6
Publisher site
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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.

Journal

Applied Mathematics & Optimization – Springer Journals

Published: Oct 1, 2023

Keywords: Viscosity solution; Fully nonlinear partial differential equations; Obliquely reflected diffusion; Non-smooth time-dependent domain; Generalized Backward stochastic differential systems; Optimal switching; 49L25; 60J50; 60H30; 60J60

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