Euler-Maruyama Scheme

The Euler-Maruyama scheme is the numerical method for integrating stochastic differential equations (SDEs). It is the stochastic analogue of the Explicit Euler method.

General Form

For the SDE $dX=f(X,t)dt+g(X,t)dW$:

$$X_{i+1}=X_i+f(X_i,t_i)dt+g(X_i,t_i)\sqrt{dt}\cdot Z_i$$

where $Z_i\sim N(0,1)$ are independent standard normal random variables drawn at each step.

Specific Cases

Wiener Process ($dW=dW$)

$$W_{i+1}=W_i+\sqrt{dt}\cdot Z$$

Ornstein-Uhlenbeck Process ($dV=-kVdt+dW$)

$$V_{i+1}=(1-kdt)V_i+\sqrt{dt}\cdot Z$$

Python

Python

import numpy as np def euler_maruyama_ou( initial_velocity: float, mean_reversion_rate: float, time_end: float, number_of_steps: int, seed: int = 0, ) -> tuple[np.ndarray, np.ndarray]: """Simulate the Ornstein-Uhlenbeck process using the Euler-Maruyama scheme. Args: initial_velocity: Starting value V(0). mean_reversion_rate: Rate k > 0 controlling mean reversion strength. time_end: End time T. number_of_steps: Number of time steps. seed: Random seed for reproducibility. Returns: Tuple of (time_array, velocity_array). """ rng = np.random.default_rng(seed) dt = time_end / number_of_steps time_array = np.linspace(0, time_end, number_of_steps + 1) velocity_array = np.empty(number_of_steps + 1) velocity_array[0] = initial_velocity for step_index in range(number_of_steps): noise = rng.standard_normal() velocity_array[step_index + 1] = ( (1 - mean_reversion_rate * dt) * velocity_array[step_index] + np.sqrt(dt) * noise ) return time_array, velocity_array

Output

Stochastic differential equation | Wiener process | Ornstein-Uhlenbeck process | Explicit Euler method | First-passage time