How does the Euler–Maruyama method compare to more advanced solvers?

The Euler–Maruyama method is a straightforward numerical approach for solving stochastic differential equations (SDEs) but is generally less accurate and stable compared to advanced solvers. It extends the deterministic Euler method to handle stochastic terms by approximating the solution incrementally using a simple discretization of noise. While easy to implement, it has limitations in convergence speed and error control. More advanced methods, such as the Milstein scheme or stochastic Runge–Kutta variants, offer better accuracy and stability by accounting for higher-order terms or using adaptive step sizes. The choice between them depends on the problem’s complexity, required precision, and computational constraints.

A key difference lies in convergence order. Euler–Maruyama has a strong convergence order of 0.5 and weak order of 1, meaning its error scales with the square root of the step size for pathwise accuracy. In contrast, the Milstein method achieves strong order 1 by including a correction term for the noise’s interaction with the system’s drift, significantly reducing error for systems with multiplicative noise (where noise depends on the state). For example, in modeling asset prices with geometric Brownian motion, Milstein’s correction term explicitly handles the state-dependent noise, leading to more accurate simulations than Euler–Maruyama. Advanced solvers like implicit methods or those with adaptive step sizing (e.g., the Kloeden–Platen method) further improve stability for stiff systems or highly nonlinear SDEs, where Euler–Maruyama might diverge or require impractically small steps.

Practical trade-offs matter. Euler–Maruyama’s simplicity makes it ideal for prototyping or problems where quick implementation outweighs precision needs. However, advanced methods are preferable when accuracy or efficiency is critical. For instance, simulating chemical reactions with fast dynamics might require an adaptive solver to handle varying timescales, while Euler–Maruyama’s fixed step size could miss rapid transitions. Similarly, systems with strong noise-drift coupling (e.g., mechanical systems under random vibrations) benefit from Milstein’s corrections. Developers should weigh computational cost: advanced methods often require more function evaluations or Jacobian calculations, which can offset their accuracy gains for large-scale or real-time applications. In such cases, Euler–Maruyama remains a pragmatic choice despite its limitations.