Spectral solution of stochastic Hamilton–Jacobi–Bellman equations via hybrid fractional Gegenbauer spectral collocation

This paper presents a novel hybrid spectral collocation framework for the numerical solution of stochastic Hamilton–Jacobi–Bellman (HJB) equations arising in optimal control theory under uncertainty. The proposed method utilizes shifted Gegenbauer polynomials as spatial basis functions and fractional-order Gegenbauer polynomials as temporal basis functions. This combination allows for high-order accuracy in space while effectively capturing memory effects and nonsmooth temporal dynamics induced by stochastic noise through the fractional parameter. The nonlinear Hamiltonian, derived from the stochastic differential equation via Itô’s lemma, is handled via a Newton–Raphson iterative scheme within the collocation framework. A rigorous convergence analysis is established, proving exponential convergence for smooth value functions and algebraic convergence for problems with singularities arising from degenerate diffusion. Comprehensive numerical experiments, including a stochastic resource extraction model, validate the theoretical predictions, demonstrating absolute errors ranging from [Formula: see text] to [Formula: see text] and confirming the method’s robustness compared to standard Chebyshev–Legendre approaches in handling stochastic diffusion terms.