Extend the diffusion-factor results to multidimensional stochastic factors

Extend the infinite-horizon optimal investment and consumption problem with power utility from the case where the stochastic factor Y is a one-dimensional Itô diffusion to the case where Y is a multidimensional Itô diffusion. Specifically, develop the corresponding Hamilton–Jacobi–Bellman formulation and establish existence of positive solutions along with verification of optimality for the associated controls in the multidimensional setting.

Background

Throughout the paper, the stochastic factor Y is assumed to be a one-dimensional Itô diffusion when treating the diffusion setting; this leads to a second-order ordinary differential equation (ODE) HJB characterization and a sub-/supersolution framework on open intervals.

The authors note that moving beyond one dimension would entail handling the Hamilton–Jacobi–Bellman equation as a semilinear elliptic partial differential equation on open domains, which is substantially more challenging. They explicitly state that addressing the multidimensional case is not covered and is deferred to future research, underscoring it as an unresolved direction.