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for the standard cases the answer is well known, but somehow I couldn't derive the answer for the following setting:

Suppose that $X_t = \sigma B_t + ct$, where $B$ is a Brownian motion, $c>0$ is a constant and $X_0=0$. We define $H_a = \inf \{ t: X_t =a \}$ for $\underline{a <0}$ as the first hitting time. What is the density of $H_a$ in this case?

It is different from the standard case because now

$$[H_a \leq t] = \left[ \inf_{s \leq t} X_s \leq a \right] \tag{1}$$

Thanks!

Serb
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    Previously answered here http://math.stackexchange.com/questions/1053294/density-of-first-hitting-time-of-brownian-motion-with-drift – stochasticboy321 Aug 18 '16 at 09:49
  • @stochasticboy321 Not exactly, the derivation there is not valid for $a < 0$, as far as I understand it. – Serb Aug 18 '16 at 11:49
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    Yeah, and replacing $B$ by this other Brownian motion $-B$ makes these two questions equivalent. – Did Aug 18 '16 at 14:53

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