Examples
- Any stopping time is a hitting time for a properly chosen process and target set. This follows from the converse of the Début theorem (Fischer, 2013).
- Let B denote standard Brownian motion on the real line R starting at the origin. Then the hitting time τA satisfies the measurability requirements to be a stopping time for every Borel measurable set A ⊆ R.
- For B as above, let denote the first exit time for the interval (−r, r), i.e. the first hit time for (−∞, −r] ∪ [r, +∞). Then the expected value and variance of satisfy
- For B as above, the time of hitting a single point (different from the starting point 0) has the Lévy distribution.
Read more about this topic: Hitting Time
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (16881744)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)