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sde.py
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sde.py
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#-*- encoding: utf-8 -*-
"""
solve stochastic differential equations
Copyright (c) 2016 @myuuuuun
Released under the MIT license.
"""
import math
import numpy as np
import pandas as pd
import ode
import matplotlib as mpl
import matplotlib.pyplot as plt
import matplotlib.cm as cm
np.set_printoptions(precision=3)
np.set_printoptions(linewidth=400)
pd.set_option('display.max_columns', 130)
pd.set_option('display.width', 800)
plt.rcParams['font.size'] = 13
# 日本語対応
mpl.rcParams['font.family'] = 'Osaka'
# Explicit Euler Method
# 陽的（前進）オイラー法
# 現状1階の問題のみ対応
def sde_euler(func, init, t_start, step, repeat, random_coef, random_process):
dim = 1
path = np.zeros((dim+1, repeat), dtype=float)
path[0, 0] = t_start
path[1, 0] = init
for i in range(1, repeat):
current = path[1, i-1]
path[0, i] = t_start + i * step
path[1, i] = current + func(current) * step + random_coef(current) * random_process(current, step)
return path
# Modified Euler Method
# 修正オイラー法
# 現状1階の問題のみ対応
def sde_modified_euler(func, init, t_start, step, repeat, random_coef, random_process):
dim = 1
path = np.zeros((dim+1, repeat), dtype=float)
path[0, 0] = t_start
path[1, 0] = init
k1 = 0
k2 = 0
for i in range(1, repeat):
current = path[1, i-1]
path[0, i] = t_start + i * step
rp = random_process(current, step)
# k1
k1 = func(current) * step + random_coef(current) * rp
# k2
k2 = func(current + k1*step) * step + random_coef(current + k1*step) * rp
path[1, i] = current + (k1 + k2) / 2
return path
# Explicit RK4 Method
# 4段4次ルンゲ・クッタ
# 現状1階の問題のみ対応
def sde_runge_kutta(func, init, t_start, step, repeat, random_coef, random_process):
dim = 1
path = np.zeros((dim+1, repeat), dtype=float)
path[0, 0] = t_start
path[1, 0] = init
k1, k2, k3, k4 = 0, 0, 0, 0
for i in range(1, repeat):
current = path[1, i-1]
path[0, i] = t_start + i * step
rp = random_process(current, step)
# k1
k1 = func(current) * step + random_coef(current) * rp
# k2
k2 = func(current + k1*step*0.5) * step + random_coef(current + k1*step*0.5) * rp
# k3
k3 = func(current + k2*step*0.5) * step + random_coef(current + k2*step*0.5) * rp
# k4
k4 = func(current + k3*step) * step + random_coef(current + k3*step) * rp
path[1:, i] = current + (k1 + 2*k2 + 2*k3 + k4) / 6
return path
if __name__ == '__main__':
"""
Sample: solve x'(t) = x(2-x) dt + x dW(t) where W(t) is Wiener process
analytic solution of
x'(t) = x(2-x) dt, x(0) = 1
is x(t) = 2*exp(2t) / (1 + exp(2t))
"""
func = lambda x: 2*x - x**2
init = 1
t_start = 0
step = 0.01
repeat = 200
seed = 198
ts = np.arange(t_start, step*repeat, step)
analytic_path = 2*np.exp(2*ts) / (1 + np.exp(2*ts))
approx_path = ode.euler(func, init, t_start, step, repeat)
sample_size = 100
# Euler Method
rs = np.random.RandomState(seed)
random_coef = lambda x: x
random_process = lambda x, step: rs.normal(loc=0, scale=math.sqrt(step))
euler_path = np.zeros((sample_size, repeat))
for i in range(sample_size):
euler_path[i] = sde_euler(func, init, t_start, step, repeat, random_coef, random_process)[1]
euler_average = np.zeros(repeat)
for i in range(repeat):
euler_average[i] = euler_path[:, i].mean()
# Modified Euler Method
rs = np.random.RandomState(seed)
random_coef = lambda x: x
random_process = lambda x, step: rs.normal(loc=0, scale=math.sqrt(step))
modified_euler_path = np.zeros((sample_size, repeat))
for i in range(sample_size):
modified_euler_path[i] = sde_modified_euler(func, init, t_start, step, repeat, random_coef, random_process)[1]
modified_euler_average = np.zeros(repeat)
for i in range(repeat):
modified_euler_average[i] = modified_euler_path[:, i].mean()
# RK4 Method
rs = np.random.RandomState(seed)
random_coef = lambda x: x
random_process = lambda x, step: rs.normal(loc=0, scale=math.sqrt(step))
rk4_path = np.zeros((sample_size, repeat))
for i in range(sample_size):
rk4_path[i] = sde_runge_kutta(func, init, t_start, step, repeat, random_coef, random_process)[1]
rk4_average = np.zeros(repeat)
for i in range(repeat):
rk4_average[i] = rk4_path[:, i].mean()
fig, ax = plt.subplots(figsize=(16, 8))
plt.title(r"SDE: $x'(t) = x(2-x) dt + x dW(t),\ x(0)=1,\ \Delta t = {0}$".format(step))
plt.xlabel("t")
plt.ylabel("x")
for i in range(sample_size):
plt.plot(ts, rk4_path[i], color='#cccccc', linewidth=1)
plt.plot(approx_path[0], approx_path[1], color='green', linewidth=3, label=r"Euler approx($x'(t) = x(2-x) dt$)")
plt.plot(ts, analytic_path, color='orange', linewidth=1, label=r"Analytic sol($x(t) = 2e^{2t} / (1+e^{2t}})$)")
plt.plot(ts, euler_average, color='red', linewidth=2, label="Euler mean({0}times)".format(sample_size))
plt.plot(ts, modified_euler_average, color='purple', linewidth=2, label=r"Modified Euler mean({0}times)".format(sample_size))
plt.plot(ts, rk4_average, color='black', linewidth=2, label=r"RK4 mean({0}times)".format(sample_size))
plt.ylim(0, 8)
plt.legend()
plt.show()