The Euler–Maruyama method is a straightforward and widely used numerical technique for solving stochastic differential equations (SDEs). Essentially, it extends the Euler method, which is used for deterministic ordinary differential equations, to accommodate stochastic components. The method applies a simple iterative process to approximate the solution by discretizing both the time and the stochastic processes involved. While it is easy to implement and understand, its accuracy can be limited, especially for complex SDEs or over long time intervals.
One of the main limitations of the Euler–Maruyama method is its first-order accuracy, which means that the error decreases linearly with smaller time steps. This can be insufficient for applications requiring high precision. More advanced solvers, such as the Milstein method or implicit methods, offer higher-order accuracy. For example, the Milstein method improves upon Euler–Maruyama by incorporating an additional term that accounts for the stochastic influence of the process, leading to better convergence properties. In contrast, implicit methods—like the implicit Euler–Maruyama—can be more stable for stiff equations, making them preferable in scenarios where the solution can exhibit rapid changes.
In terms of computational efficiency, the Euler–Maruyama method is typically faster due to its simplicity, which is an advantage in scenarios where speed is crucial, such as real-time simulations. However, when precision is a priority or when dealing with stiff SDEs, developers may prefer more sophisticated methods despite the increased computational cost. Overall, choosing between the Euler–Maruyama method and more advanced solvers must be based on the specific requirements of the problem at hand, such as the desired accuracy, stability, and computational resources available.