Optimal Control of Stochastic Homogeneous Systems

Abstract. This paper investigates a new class of homogeneous stochastic control problems with cone control constraints, extending the classical homogeneous stochastic linear-quadratic (LQ) framework to encompass nonlinear system dynamics and nonquadratic cost functionals. We demonstrate that, analogous to the LQ case, the optimal controls and value functions for these generalized problems are intimately connected to a novel class of highly nonlinear backward stochastic differential equations (BSDEs). We establish the existence and uniqueness of solutions to these BSDEs under three distinct sets of conditions, employing techniques such as truncation functions and logarithmic transformations. Furthermore, we derive explicit feedback representations for the optimal controls and value functions in terms of the solutions to these BSDEs, supported by rigorous verification arguments. Our general solvability conditions allow us to recover many known results for homogeneous LQ problems, including both standard and singular cases, as special instances of our framework.