I'm looking for functions $g: \mathbb{R}_+ \to \mathbb{R}$ such that the hitting time

$$\tau := \inf \{t \geq 0 : B_t \nleq g(t) \} $$

has an explicit density with respect to the Lebesgue measure, where $B_t$ is a standard Brownian motion. What are possible examples?

The case where, for some $R> 0$, $g(t) \equiv R$ is constant leads to the hitting time having Lévy distribution. Indeed for $g(t) = R - \alpha t$, we also obtain an explicit density of $$\rho_{\tau}(t) = \frac{R}{\sqrt{4\pi t^3}} e^{-\frac{(R - \alpha t)^2}{4t}}$$

Are there other functions out there where a similar representation exists? Is there an explicit density for, say, $\inf \{t \geq 0 : B_t \nleq M_t \}$ for some process $M_t$ with $M_0 > 0$.

  • $\begingroup$ The difference between two Brownian motions is a Brownian motion, so your last question is a special case of your first $g(t) \equiv R$. $\endgroup$ – Timothy Budd Dec 12 '19 at 16:41
  • $\begingroup$ You're absolutely correct, of course. I'll change the question to a more general wording, but that's very helpful, I'm not sure how I did not think of that. $\endgroup$ – Marc Vaisband Dec 12 '19 at 17:13

See e.g. ANALYTIC CROSSING PROBABILITIES FOR CERTAIN BARRIERS BY BROWNIAN MOTION, especially Example 2 (page 9) there for a square-root barrier, Example 3 (page 9) there for a square-root-times-log-factor barrier, and further references therein.

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