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Brownian motion with drift

{title2=$T=\sup\{t\geq0:W_t-at>b\}$}

For $a,b>0$ and the convention $\sup\varnothing=0$,
$$
\mathbb P(T\leq t)
=\Phi\!\left(a\sqrt t+\frac b{\sqrt t}\right)
-e^{-2ab}\Phi\!\left(\frac b{\sqrt t}-a\sqrt t\right),
\qquad t>0.
$$


Last passage time above a level for Brownian motion with negative drift

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 29K / d / Solution

Last passage time above a level for Brownian motion with negative drift ( $T = sup {t \geq 0 : W_{t} - a t > b}$ )

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Last passage time above a level for Brownian motion with negative drift (T=sup{t≥0:Wt​−at>b})

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Last passage time above a level for Brownian motion with negative drift ( $T = sup {t \geq 0 : W_{t} - a t > b}$ )

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