Combinatorial Approximation Method for the Fractional Stochastic Hamilton–Jacobi–Bellman Equation

We introduce a combinatorial method for approximating the solution of a very complicated nonlinear fractional stochastic partial differential equation (SPDE) which appears in optimal stochastic control. We extend our previous research on the fractional SABR (Stochastic Alpha Beta Rho) model where we could derive only an approximation of the shadow price without the explicit formulas for utility function maximization. We aim to solve the equation by integrating combinatorial techniques with fractional calculus to address the system’s inherent randomness and memory effects. The ensuing approximation framework provides analytical tractability for the fractional stochastic Hamilton–Jacobi–Bellman equation and shows promise of applicability to fields like quantitative finance, physics, and engineering, where sound decisionmaking under uncertainty is critical.