Summary

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 }}

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