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If you have a continuous stochastic process, then how is the hitting time for a closed set different from the hitting time of an open set when we're trying to show it is a stopping time?

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See the related discussion here. Please tell if it is what you wanted - or your question is more specific – Ilya Feb 22 '12 at 11:55

Some examples to consider:

  1. with a standard Wiener process, the hitting time for non-negative values (closed) is zero, but for positive values (open) need not be.

  2. for a stochastic process restricted to a corridor, there may be a hitting time for hitting the boundaries of the corridor (closed), but not for passing beyond them (open).

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