First-Passage Time

The first-passage time $\tau$ of a stochastic process $X(t)$ to a threshold $b$ is defined as the first time the process reaches or exceeds $b$:

$$\tau =\inf \{t\ge 0:X(t)\ge b\}$$

Numerical Method

To estimate the mean first-passage time numerically:

1. Run an Euler-Maruyama scheme simulation of the process (e.g. the Ornstein-Uhlenbeck process).
2. At each time step, check whether $X\ge b$.
3. When the condition is met, record the elapsed time $t$ as $\tau$ for that walker.
4. Average $\tau$ over many independent simulations (walkers) to obtain the mean first-passage time $⟨\tau ⟩$.

Python

Python

import numpy as np def first_passage_time_ou( initial_velocity: float, mean_reversion_rate: float, threshold: float, time_step: float, max_time: float, seed: int = 0, ) -> float | None: """Estimate the first-passage time for a single OU walker. Args: initial_velocity: Starting value V(0). mean_reversion_rate: Rate k > 0. threshold: Threshold b; simulation stops when V >= b. time_step: Time step dt for the Euler-Maruyama scheme. max_time: Maximum simulation time before giving up. seed: Random seed for reproducibility. Returns: First-passage time tau, or None if threshold not reached within max_time. """ rng = np.random.default_rng(seed) velocity = initial_velocity current_time = 0.0 while current_time < max_time: if velocity >= threshold: return current_time velocity = ( (1 - mean_reversion_rate * time_step) * velocity + np.sqrt(time_step) * rng.standard_normal() ) current_time += time_step return None # Example: OU process with k=3.0, threshold=1.35 mean_reversion_rate = 3.0 threshold = 1.35 time_step = 0.001 max_time = 1000.0 tau = first_passage_time_ou( initial_velocity=-0.5, mean_reversion_rate=mean_reversion_rate, threshold=threshold, time_step=time_step, max_time=max_time, seed=0, ) if tau is not None: print(f"tau = {tau:.3f}")

Output

Euler-Maruyama scheme | Ornstein-Uhlenbeck process | Langevin equation | Wiener process | Stochastic differential equation