Take a standard stochastic difference equation, $$ d z(t) = \gamma dt + \sigma d W(t) $$ with $W(t)$ standard brownian motion.

Take an initial condition $z(0) = z_0$ and a threshold $\underline{z} < z_0$. Define the stopping time $T$ as the first time that $z(t) \leq \underline{z}$.

Question: What is the survivor function of the stopping time? Alternatively, you could give me the CDF and/or PDF.

Note: This seems like such a standard problem, that I thought this would have an easily found formula. e.g. a variation on the Bachelier-Levy formula, or the corollary 7.2.2 in Shreve's stochastic calculus for finance.

Certainly plenty of versions with drift exist. However, I cannot find a variation of the formula with the variance $\sigma$. I would rather have a reference to the formula to ensure I don't make a mistake in any derivation.

  • 1
    $\begingroup$ $T$ follows an inverse Gaussian distribution $\endgroup$ Aug 29, 2016 at 19:54
  • $\begingroup$ @JarleTufto Perfect, thanks. Post a solution so that I can accept it. $\endgroup$
    – jlperla
    Aug 29, 2016 at 20:05
  • 1
    $\begingroup$ I don't remember the details but I think a derivation can be found in Karlin & Taylor $\endgroup$ Aug 29, 2016 at 20:15
  • $\begingroup$ This thread has a derivation using the "reflection principle" for Brownian motion without drift. $\endgroup$ Aug 30, 2016 at 15:38
  • $\begingroup$ Found it: Karlin Taylor, first course. Theorem 5.3. Thanks! $\endgroup$
    – jlperla
    Aug 30, 2016 at 21:32


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