# Explicit densities for Brownian motion hitting times

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 $$M_0 > 0$$.

• The difference between two Brownian motions is a Brownian motion, so your last question is a special case of your first $g(t) \equiv R$. Dec 12 '19 at 16:41
• 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. Dec 12 '19 at 17:13