New Numerical Methods for Solving Systems of Stochastic Differential Equations

Abstract

This study investigates and compares the numerical performance of the Euler–Maruyama and Hybrid RK–Milstein methods for solving stochastic differential equations (SDEs). Despite the wide use of Euler–Maruyama for its simplicity, its accuracy and stability degrade under strong stochastic effects, highlighting the need for a more stable hybrid approach. The methodology involves implementing both algorithms in MATLAB using multiple time steps (Δt = 0.0078125 to 0.0009766) and evaluating their strong convergence, runtime efficiency, and stability. A reference solution with 8192-time steps was used for benchmarking, and both statistical and graphical analyses were performed to assess performance. The results show that Euler–Maruyama achieved a strong convergence rate of approximately −0.56 with lower runtime (0.33–1.95 s), while the Hybrid RK–Milstein method maintained stable errors (~0.045) but required higher computational time (2.77–15.37 s). The hybrid method produced smoother trajectories and better variance control, especially in high-noise conditions. For future work, adaptive hybrid algorithms are proposed to balance computational cost and numerical stability for complex long-term stochastic simulations.