System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains

Manal Jakani (1, 2)

1 LMM - Laboratoire Manceau de Mathématiques
2 LIBMA - Laboratoire Ibn Al-Banna de Mathématiques et Applications

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 Hamilton-Jacobi-Bellman 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 PDEs system.

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

Domaines: Mathématiques [math]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Manal Jakani. System of nonlinear second-order parabolic partial differential equations with interconnected obstacles and oblique derivative boundary conditions on non-smooth time-dependent domains. 2022. ⟨hal-03913755⟩