Visualization of Brownian bridge construction of a Brownian motion

brownian_bridge_construction.py

"""Visualization of Brownian bridge construction of a Brownian motion."""

import numpy as np
from matplotlib import pyplot as plt


T = 1
TAB10_CMAP = plt.get_cmap("tab10")
DEFAULT_COLOR = TAB10_CMAP(0)


def generate_brownian_path(num_steps, level=0):
    np.random.seed(level)
    if level == 0:
        time = np.arange(num_steps + 1) / num_steps
        eps = np.concatenate([np.zeros(1), np.random.randn(num_steps)])
        path = np.cumsum(eps) / np.sqrt(num_steps)
    elif level > 0:
        time = np.arange(num_steps * 2 ** level + 1) / (num_steps * 2 ** level)
        _, path0 = generate_brownian_path(num_steps, level - 1)
        eps = np.random.randn(num_steps * 2 ** (level - 1))
        path = np.empty((num_steps * 2 ** level + 1,))
        path[::2] = path0
        path[1::2] = (path0[:-1] + path0[1:]) / 2 + eps / np.sqrt(2 * num_steps * 2 ** level)
    else:
        raise ValueError("level must be non-negative integer.")
    return time, path


def plot_bridge_construction(max_level=5):
    for level in range(max_level):
        time, path = generate_brownian_path(1, level)
        if level == 0:
            plot_axis()
            plt.plot(time, path, c=DEFAULT_COLOR, alpha=1 / max_level)
        elif level > 0:
            plt.plot(time, path, linestyle='-', c=DEFAULT_COLOR, alpha=(level + 1) / max_level)
        else:
            raise ValueError("level must be non-negative integer.", c=0)

    plt.xlabel('Time t')
    plt.ylabel('Path')
    plt.xlim([-0.2, 1.1])
    plt.ylim([-0.5, 3.0])


def plot_axis():
    ax = plt.gca()
    ax.axis("off")
    plt.axhline(0,color='black') # x = 0
    plt.axvline(0,color='black') # y = 0


if __name__ == "__main__":
    plt.figure(figsize=(6, 18))
    for i in range(10):
        plt.subplot(5, 2, i + 1)
        plot_bridge_construction(i + 1)
    plt.show()

This script generates a figure like this: