兩個獨立的 OU 程序同時超過障礙的機率是多少？

我有兩個 OU 程序。

我想知道在 0 到 T 的時間段內它們至少同時超過障礙一次的機率。

這是我的 Python 程式碼，用於提示您的問題。它模擬 Ornstein Uhlenbeck 過程併計算接觸障礙物的機率 $ B < r_0 $ . 應該針對兩個獨立OU的這種情況進行調整。

import numpy as np
from matplotlib import pyplot as plt
from math import *
import numpy.random as nrand

plt.style.use('seaborn')

# Parameters

r0 = 0.02  # Starting interest rate
time = 100  # Simulation time
delta = 1 / 252  # Delta time
sigma = 0.03  # Volatility
ou_a = 20  # Rate of mean reversion
ou_mu = 0.02  # Long run average
end = 90  # Time = T
end = end + 1
i = 1000  # Number of simulations
barrier = 0.01  # Level of barrier


def brownian_motion_log_returns(dt, sig):
   sqrt_delta_sigma = sqrt(dt) * sig
   return nrand.normal(loc=0, scale=sqrt_delta_sigma, size=time)


def ornstein_uhlenbeck(t, a, mu, dt):
   paths = [r0]
   brownian_motion_returns = brownian_motion_log_returns(delta, sigma)
   for i in range(1, t):
       drift = a * (mu - paths[i - 1]) * dt
       randomness = brownian_motion_returns[i - 1]
       paths.append(paths[i - 1] + drift + randomness)
   return paths


# MONTE CARLO SIMULATION

paths = []
for i in range(i):
   level = ornstein_uhlenbeck(time, ou_a, ou_mu, delta)
   paths.append(level)

paths = np.asarray(paths)
paths = paths.T
new_paths = np.delete(paths, np.s_[end:], 0)  # Remove paths beyond T
print(np.shape(new_paths))

# MC projection graph

plt.plot(new_paths, lw=0.5)
plt.axhline(y=barrier, color='b', linestyle='-', lw=0.7)
plt.title('Monte Carlo Simulations')
plt.xlabel('Time')
plt.ylabel('Level')
plt.show()


def prob(path, b, i):
   count = 0
   for col in path.T:
       a = min(col)
       if a < b:
           count = count + 1
   p = count / i
   return p


mcp = prob(new_paths, barrier, i)
print('Monte Carlo probability of touching:', mcp)

引用自：https://quant.stackexchange.com/questions/38568