模拟布朗运动

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
plt.rcParams['font.family'] = ['sans-serif']
plt.rcParams['font.sans-serif'] = ['SimHei']
import warnings
warnings.filterwarnings("ignore")

plt.figure(figsize=(10, 4.5))
delta_t = 0.01
rd = np.cumsum(np.sqrt(delta_t) * np.random.randn(999))
rd = np.insert(rd, 0, 0)
t = np.arange(0, 10, 0.01)
plt.plot(t, rd);plt.title('标准布朗运动的轨道')
plt.show()

plt.figure(figsize=(11, 17))
for i in range(4):
    plt.subplot(4, 1, i+1)
    mu, sigma = 0.1, 0.5
    t = np.arange(0, 10, 0.01)
    delta_t = 0.01
    path = [10]
    for i in range(1, t.size):
        path.append(path[-1] + mu * path[-1] * delta_t + \
            sigma * path[-1] * np.sqrt(delta_t) * np.random.randn())
    plt.plot(t, path, label=f'$\mu={mu}, \sigma={sigma}$')
plt.suptitle(f'$S_0=10, \mu={mu}, \sigma={sigma}$的几何布朗运动的轨道', y=0.91); plt.show()

plt.figure(figsize=(11, 17))
for i in range(4):
    plt.subplot(4, 1, i+1)
    mu, sigma = 0.5, 0.8
    t = np.arange(0, 10, 0.01)
    delta_t = 0.01
    path = [10]
    for i in range(1, t.size):
        path.append(path[-1] + mu * path[-1] * delta_t + \
            sigma * path[-1] * np.sqrt(delta_t) * np.random.randn())
    plt.plot(t, path, label=f'$\mu={mu}, \sigma={sigma}$')
plt.suptitle(f'$S_0=10, \mu={mu}, \sigma={sigma}$的几何布朗运动的轨道', y=0.91); plt.show()

mu, sigma = 0.2, 0.8
t = np.arange(0, 1, 0.01)
delta_t = 0.01
distribution = []
for _ in range(10000):
    path = [10]
    for i in range(1, t.size):
        path.append(path[-1] + mu * path[-1] * delta_t + \
            sigma * path[-1] * np.sqrt(delta_t) * np.random.randn())
    distribution.append(path[-1])

sns.distplot(distribution, kde=False, norm_hist=True)
plt.title(f'$\mu={mu}, \sigma={sigma}, S_1$的分布')
plt.show()