DescriptionIto lemma 1D illustration.svg	English: ```python import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm def f(x): return (x-1)**2 + 1*(x-1)+0.5 mean = 1 xs = np.random.normal(mean, 0.1, 2000) ys = f(xs) for x, y in zip(xs[:200], ys[:200]): plt.plot([x, x], [0, y], 'g', alpha=0.1) plt.plot([0, x], [y, y], 'g', alpha=0.1) plt.plot([mean, mean], [0, f(mean)], 'r') plt.plot([0, mean], [f(mean), f(mean)], 'r') x_values = np.linspace(0, 1.2, 100) plt.plot(x_values, f(x_values), label=r'f', color='b') plt.xlabel(r'$X_{t+dt} = X_t + \mu_t dt + z\sigma_t\sqrt{dt }) plt.ylabel(r'$f(X_{t+dt})
Source	Own work
Author	This file is lacking author information.

)

params_x = norm.fit(xs) params_y = norm.fit(ys)

1. Create Gaussian distributions for xs and ys

x_dist = norm(loc=params_x[0], scale=params_x[1]) y_dist = norm(loc=params_y[0], scale=params_y[1])

1. Plot the Gaussian distribution over xs as (x, rho_x(x))

x_values = np.linspace(x_dist.mean() - 3*x_dist.std(), x_dist.mean() + 3*x_dist.std(), 100) rho_x = x_dist.pdf(x_values) plt.plot(x_values, rho_x/max(rho_x)/4, label=r'distribution of $X_{t+dt}}}, color='k')

1. Plot the Gaussian distribution over ys as (rho_y(y), y)

y_values = np.linspace(y_dist.mean() - 3*y_dist.std(), y_dist.mean() + 3*y_dist.std(), 100) rho_y = y_dist.pdf(y_values) plt.plot(rho_y/max(rho_y)/4, y_values, label=r'distribution of $f(X_{t+dt})}}, color='k')

plt.legend(loc='upper left')

plt.grid() plt.savefig("ito_lemma.svg") plt.show() ```}} |date=2023-10-17 |source=Own work |author=Cosmia Nebula }}

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue