1 What is a First Passage Time?
A first passage time (also called a hitting time) is the first time at which a stochastic process reaches or crosses a specified threshold or set. It is a canonical example of a stopping time.
Formally, given a stochastic process \((X_n)\) and a target set \(A\), the first passage time is: \[ \tau_A = \inf\{n \geq 0 : X_n \in A\} \]
The infimum is \(\infty\) if the process never reaches \(A\).
2 Examples
2.1 Random Walk
For a simple symmetric random walk \((S_n)\) starting at 0, the first time it hits level \(k > 0\) is: \[ \tau_k = \inf\{n \geq 1 : S_n = k\} \]
This is a stopping time because we can determine at time \(n\) whether \(\tau_k \leq n\) based only on the path up to time \(n\).
2.2 Barrier Crossing
In finance, the first time a stock price \(P_t\) hits a barrier level \(B\) is a hitting time commonly used in option pricing and barrier option analysis.
2.3 Gambler’s Ruin
A gambler with initial wealth \(x\) plays until reaching \(0\) (ruin) or \(N\) (goal). The first time either occurs is \(\tau = \inf\{n : X_n = 0 \text{ or } X_n = N\}\).
3 Properties
First passage times are stopping times: By definition, \(\{\tau \leq n\} \in \mathcal{F}_n\) since we need only observe the process up to time \(n\) to determine whether we’ve reached \(A\) by then.
Hitting probability: For a process starting at \(x\), define \(P_x(A)\) as the probability of eventually reaching set \(A\): \[ P_x(A) = \mathbb{P}_x(\tau_A < \infty) \]
4 Applications
- Structural Breaks: Detection of when a time series changes regime
- Sequential Decision-Making: Optimal stopping when to act based on threshold crossing
- Failure Analysis: Time until a system fails (reliability theory)
- Brownian Motion: First passage time of Brownian motion and its variants (e.g., hitting time above a barrier)
- Queuing Theory: Time until queue reaches capacity