Comparative Analysis of Metaheuristic Optimization Techniques for Solving Nonlinear Fractional Riccati Stochastic Differential Equations

Mathematically and computationally, issues arise in the modeling and simulation of nonlinear fractional-order Riccati Stochastic Differential Equations (RSDEs) as a result of their memory dependence, non-linearity, and stochasticity. Such equations are critical in many fields of application, including control theory and financial mathematics, but their numerical solution is little studied. This gap is investigated in this research through this effort in the development of a very robust hybrid computational method to tackle the solutions of fractional-order RSDEs. The contribution to the research is the development and application of a comparative optimization based method which combines three metaheuristic algorithms (Particle Swarm Optimization (PSO), Firefly Algorithm (FFA), and Artificial Bee Colony (ABC)) to enact precise nonlinear RSDE that are characterized by Caputo-type fractional derivatives. The suggested algorithm minimizes an uncontrolled fitness feature simulation with Brownian incremental steps and numerical changes of the fractional derivation. Parameters of trial solutions, each algorithm minimizes the residual error of the RSDE within a discretized time grid. The numerical experiments prove that the PSO-based framework obtained better precision, accelerates, and has better numerical stability over the time horizon FFA, ABC, and standard methods of solution. The fair comparison of all the algorithms is carried by the same stochastic realization of the Brown path. Detailed analyses of the error, like mean squared errors, and absolute errors, revealed that the PSO is efficient to capture both deterministic and stochastic nature of the RSDEs. To sum up, the suggested PSO-based metaheuristic algorithm framework offers a very successful and broadly applicable approach to the numerical approxima tion of fractional stochastic dynamics, and hence develops the repertoire of computational scheme accessible to model complex systems in the sciences and engineering.