Distribution of the first passage time of a Gaussian random walk Does anyone know the distribution for the first passage time of a Gaussian random walk i.e.
$$
S_n = \sum_{i=1}^n X_i
$$
where $X_i$ are iid normally distributed random variables. The first passage time is
$$
\tau = inf\{n: S_n \geq C\}
$$
where $C$ is a constant. The literature I have come across mostly deal with expectations and even then is more focused on trying to bound the expectations and examine the limiting behaviour as $C \rightarrow \infty.$ Really appreciate any help you can provide.
 A: For every $|z|\leqslant1$, the generating function $u(x)=E_x(z^\tau)$ for the random walk starting at $x$ is the unique solution of the integral identity
$$
u(x)=z\cdot\int_\mathbb Ru(x+y)g(y)\mathrm dy,
$$
where $g$ is the standard normal density, with the boundary condition that $u(x)=1$ for every $x\geqslant C$.
A: The random walk (in 1$d$) starting at $y>0$ at time $t=0$ and reaching $x=0$ at time $t>0$ has the following distribution:
\begin{equation}
F(x=0,t>0\big|x=y>0,t=0)=\frac{y}{\sqrt{4\pi D\,t^3}}\text{exp}\left(\frac{-y^2}{4Dt}\right)
\end{equation}
where $D$ is the diffusion coefficient. Integrating this equation from $t:0\rightarrow \infty$ gives unity. This corresponds to the fact that a random walk in 1$d$ is recurrent. A nice book on the subject is: A kinetical view of statistical physics by Sidney Redner et al.
A: Note that $P(\tau\leq n)=P(\max_{1\leq k\leq n} S_k \geq C)$. So you might want to have a look at a paper by Janssen and van Leeuwaarden: "Cumulants of the maximum of the Gaussian random walk". 
