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 
be a stochastic process, and let S be a measurable subset of the state space 
. Then the first hit time ![\tau_{S} : \Omega \to [0, + \infty]](../../../math/a/f/f/aff9edab7f59e7fe3d68436e82e93078.png)
is the random variable defined by
The first exit time (from S) is defined to be the first hit time for 
.
The first return time is defined to be the first hit time for the singleton set 
, 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 
starting at the origin. Then the hitting time τS is a stopping time for every Borel measurable set 
.
Let τr, r > 0, denote the first exit time for the interval ( − r,r), i.e. the first hit time for ![(- \infty, - r] \cup [r, + \infty)](../../../math/9/d/8/9d87cff5fe8fe0da59a0ef775d82c376.png)
. Then the expected value and variance of τr satisfy
![\mathbb{E} \left[ \tau_{r} \right] = r^{2},](../../../math/e/d/f/edfcd108fc468b3643a6957e829983f8.png)
![\mathrm{Var} \left[ \tau_{r} \right] = 2 r^{4}.](../../../math/5/4/f/54f36148b89458b72a70ac1d349c1f70.png)
