Constrained Mean Variance Portfolio Selection under Reflected Factor Dynamics: Stochastic Riccati Equations, Neumann Problems,and a Deep Solver

Abstract

We study extended stochastic Riccati equations (ESREs) arising in constrained continuous time mean variance portfolio selection with random market coefficients. Reflected Markovian diffusions are introduced to model factor dynamics in a bounded state space, naturally leading to homogeneous Neumann boundary conditions. A logarithmic transformation converts the extended stochastic Riccati equations into quadratic backward stochastic differential equations (BSDEs) driven by the reflected factors. We establish their probabilistic representation through semilinear parabolic Neumann problems and prove existence and uniqueness of bounded continuous viscosity solutions. Existence is obtained through stability and comparison arguments for quadratic BSDEs, whereas uniqueness is established by combining a Kobylanski-type transformation with a viscosity comparison principle adapted to the Neumann boundary setting. We further develop a Deep BSDE method combining reflected-path simulation, geometry-aware boundary treatment, and efficient evaluation of the constrained Hamiltonian. Numerical experiments demonstrate the effectiveness of the method, providing a mesh-free framework for high-dimensional constrained portfolio problems with Neumann boundary.