Non-Radial Solution Structures for Quasilinear Hamilton--Jacobi--Bellman Equations in Bounded Settings

Abstract: We study quasilinear Hamilton--Jacobi--Bellman equations on bounded smooth convex domains. We show that the quasilinear Hamilton--Jacobi--Bellman equations arise naturally from stochastic optimal control problems with exit-time costs. The PDE is obtained via dynamic programming applied to controlled It^{o} diffusions, providing both a probabilistic interpretation and a rigorous derivation. Our result establishes existence and uniqueness of positive classical solutions under sub-quadratic growth conditions on the source term. The constructive proofs, based on monotone iteration and barrier techniques, also provide a framework for algorithmic implementation with applications in production planning and image restoration. We provide complete detailed proofs with rigorous estimates and establish the connection to stochastic control theory through the dynamic programming principle.