brownian_motion.py

import numpy as np
np.random.seed(1234)

def brownian_motion(N, T, h):
    """
    Simulates a Brownian motion
    :param int N : the number of discrete steps
    :param int T: the number of continuous time steps
    :param float h: the variance of the increments
    """   
    dt = 1. * T/N  # the normalizing constant
    random_increments = np.random.normal(0.0, 1.0 * h, N)*np.sqrt(dt)  # the epsilon values
    brownian_motion = np.cumsum(random_increments)  # calculate the brownian motion
    brownian_motion = np.insert(brownian_motion, 0, 0.0) # insert the initial condition
    
    return brownian_motion, random_increments

N = 50 # the number of discrete steps
T = 1 # the number of continuous time steps
h = 1 # the variance of the increments
dt = 1.0 * T/N  # total number of time steps

# generate a brownian motion
X, epsilon = brownian_motion(N, T ,h)