A Hybrid Fractional Chebyshev–Legendre Spectral Collocation Method for Hamilton–Jacobi–Bellman Equations

Stochastic optimal control problems are commonly formulated as optimization problems constrained by stochastic dynamical systems, whose value functions satisfy Hamilton–Jacobi–Bellman (HJB) equations. Owing to their strong nonlinearity and high dimensionality, closed-form solutions of HJB equations are rarely available, thereby motivating the development of robust and highly accurate numerical methods. This research introduces two hybrid spectral–collocation strategies for the numerical solution of stochastic HJB equations, constructed from different combinations of orthogonal polynomial bases. The first strategy utilizes shifted Chebyshev polynomials for time approximation and fractional-order Legendre polynomials for state approximation, while the second utilizes shifted Legendre polynomials in time and fractional-order Chebyshev polynomials in state. A convergence analysis is developed within the Caputo fractional derivative framework to justify the proposed methods and to establish the associated error estimates. The resulting nonlinear algebraic system is then solved using the collocation method. Numerical simulations, including an application to a resource extraction model, confirm that the proposed methods attain a high level of accuracy and exhibit convergence rates in strong agreement with the theoretical predictions. These results demonstrate that the developed hybrid spectral–collocation frameworks constitute reliable and efficient tools for addressing stochastic optimal control problems based on HJB equations.