EulerMaruyama

Euler-Maruyama method is a numerical technique used to approximate the solution of stochastic differential equations (SDEs). It is an extension of the Euler method for ordinary differential equations (ODEs) to the stochastic case. The method was developed by Leonhard Euler and Kiyosi Itô, and it is widely used in various fields such as finance, physics, and engineering.

The Euler-Maruyama method is based on the idea of discretizing the time interval and approximating the stochastic process at each time step. The method uses the following iterative scheme:

X(t_n+1) = X(t_n) + f(t_n, X(t_n)) * Δt + σ(t_n, X(t_n)) * ΔW_n

where X(t) is the stochastic process, f(t, X(t)) is the drift term, σ(t, X(t)) is the diffusion term, Δt is the time step, and ΔW_n is the increment of a Wiener process (standard Brownian motion) over the time interval [t_n, t_n+1].

The Euler-Maruyama method is a first-order method, meaning that the local truncation error is proportional to Δt. This implies that the method converges to the true solution as Δt approaches zero, but the rate of convergence is relatively slow. However, the method is simple to implement and computationally efficient, making it a popular choice for many applications.

One of the main advantages of the Euler-Maruyama method is its ability to handle high-dimensional SDEs, which are often encountered in practice. The method can be easily extended to handle multiple stochastic processes and can be used to simulate complex systems with many interacting components.

In summary, the Euler-Maruyama method is a powerful numerical technique for approximating the solution of stochastic differential equations. It is based on the idea of discretizing the time interval and approximating the stochastic process at each time step. The method is simple to implement, computationally efficient, and can handle high-dimensional SDEs. However, it is a first-order method, meaning that the rate of convergence is relatively slow. Despite this, the method is widely used in various fields such as finance, physics, and engineering.