# What is the expected inverse stopping time for an Brownian Motion?

Let $$B_t$$ be standard Brownian motion and $$\tau_a=\inf\{t\geq 0 : B_t \geq a\}$$ be the stopping time where $$B_t$$ exceeds some value $$a$$.

Is there an analytic form for $$\mathbb{E}\left[\frac{1}{\tau_a}\right]$$ ?

Context:

Suppose $$B_t$$ represents the price of a financial instrument. If an investor buys this instrument today at a price of $$p$$ and plans to sell at a profit of $$a$$, then the expected (continually-compounded) return $$r$$ of the investor is given by

$$r = \log\left(1+\frac{a}{p}\right)\mathbb{E}\left[\frac{1}{\tau_a}\right]$$

I wish to find an analytic form for $$r$$ so that I may then maximise $$r$$ over all $$a$$. That is, I wish to find the optimal price at which the investor should sell the instrument in order to maximise her return.

First, note that for all $$t\ge0$$, we have the equality of events : $$\{\tau_a\le t\} = \left\{\sup_{0\le s \le t} B_s \ge a\right\}$$ (In words, this says that the Brownian Motion $$B$$ reaches level $$a$$ before time $$t$$ if and only if its supremum on $$[0,t]$$is at least $$a$$).
By the reflection principle and the above equality, we thus have $$\mathbb P\left(\tau_a\le t\right)=\mathbb P\left(\sup_{0\le s \le t} B_s \ge a\right)=2\mathbb P\left(B_t\ge a\right)$$ The above is true for any $$t>0$$ and in particular it is true for $$t'\equiv 1/t$$, so we have
\begin{align} F_a(t)\equiv\mathbb P\left(\frac {1}{\tau_a}\le t\right) &= \mathbb P\left(\tau_a\ge \frac 1 t\right)\\ &= 1 - \mathbb P\left(\tau_a\le \frac 1 t\right)\\ &= 1 - 2\mathbb P\left(B_{1/t}\ge a\right) \\ &= 1 - 2\mathbb P\left(tB_{1/t}\ge ta\right)\\ &= 1 - 2\mathbb P\left(B_t\ge ta\right)\tag1\\ &= 1 - 2\mathbb P \left(Z\ge a\sqrt t\right)\tag2\\ \end{align} Where I used the time inversion property in $$(1)$$ and the fact that $$Z\equiv B_t/\sqrt t\sim \mathcal N(0,1)$$ in $$(2)$$.
At this point, it is quite straightforward to express the CDF of $$\frac {1}{\tau_a}$$ $$F_{a}$$ in terms of the standard normal CDF. In particular, if you want to compute $$\mathbb E\left[\frac {1}{\tau_a}\right]$$, you have at least two options :
1. By a well-known formula, you can integrate $$1-F_a$$ directly : $$\mathbb E\left[\frac {1}{\tau_a}\right] = \int_0^\infty (1-F_a(t))\ dt = \int_0^\infty \left[1-\text{erf}\left(\frac{a\sqrt t}{\sqrt 2}\right)\right]\ dt$$
2. You can compute the PDF $$f_a(t) :=dF_a(t)/dt$$ and integrate $$\mathbb E\left[\frac {1}{\tau_a}\right] = \int_0^\infty tf_a(t)\ dt$$
Both of these integrals involve very standard functions and by digging up the literature you should be able to find a way to express them in terms of $$\text{erf}, \text{erfc}$$ or other well-known quantities, which should allow you to approximate $$\mathbb E\left[\frac {1}{\tau_a}\right]$$ well enough for your purposes.