(a) Derive the distribution of the time taken by a Brownian motion {x(t), t >= 0} to cross the level α > 0 given x(0) = 0. (b) Let T_(i) be the time that the conservative Markov process stays in state i. Show that P[T_(i) <= t] = 1 - e^(-q_(ii)) where (q_(ij)) is the generator of the Markov process. Show that the state i is absorbing if q_(it) = 0.
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To derive the distribution of the first passage time of a Brownian motion to cross the level α > 0, we can use the reflection principle. The reflection principle states that the probability of a Brownian motion hitting a level α > 0 for the first time is twice Show more…
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