Hitting time

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In the study of stochastic processes in mathematics, a hitting time (or first hit time) is a particular instance of a stopping time, the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times.

[edit] Definitions

Let $X : [0, + \infty) \times \Omega \to \mathbb{X}$ be a stochastic process, and let $S$ be a measurable subset of the state space $\mathbb{X}$ . Then the first hit time $\tau_{S} : \Omega \to [0, + \infty]$ is the random variable defined by

$\tau_{S} (\omega) := \inf \{ t > 0 | X_{t} (\omega) \in S \}.$

The first exit time (from $S$ ) is defined to be the first hit time for $\mathbb{X} \setminus S$ .

The first return time is defined to be the first hit time for the singleton set $\{ X_{0} (\omega) \} \subseteq \mathbb{X}$ , which is usually a given deterministic element of the state space, such as the origin of the coordinate system.

[edit] Example

Let $B$ denote standard Brownian motion on the real line $\mathbb{R}$ starting at the origin. Then the hitting time $τ S$ is a stopping time for every Borel measurable set $S \subseteq \mathbb{R}$ .

Let $τ r$ , $r > 0$ , denote the first exit time for the interval $( - r, r)$ , i.e. the first hit time for $(- \infty, - r] \cup [r, + \infty)$ . Then the expected value and variance of $τ r$ satisfy