Definitions
Let T be an ordered index set such as the natural numbers, N, the non-negative real numbers, is the random variable defined by
The first exit time (from A) is defined to be the first hit time for S \ A, the complement of A in S. Confusingly, this is also often denoted by τA (e.g. in Øksendal (2003)).
The first return time is defined to be the first hit time for the singleton set { X0(ω) }, which is usually a given deterministic element of the state space, such as the origin of the coordinate system.
Read more about this topic: Hitting Time
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