\title{An Open Problem: First-Passage time of Brownian Motion \\
to an Exponentially Decaying Boundary}

Let \((W_t)_{t \ge 0}\) be a standard Brownian motion with \(W_0 = 0\), and let
\(b : [0,\infty) \to \mathbb{R}_{>0}\) be the exponentially decaying boundary
\[
  b(t) = b_0 \, e^{-ct}, \qquad b_0 > 0,\ c > 0.
\]
Define the first-passage time
\[
  \tau \;=\; \inf\{\, t > 0 : W_t \ge b(t) \,\},
\]
with distribution function \(F(t) = \mathbb{P}(\tau \le t)\). We are interested in finding explicitly the distribution of \(\tau\). Beyond its theoretical interest, this problem has a direct application in computational neuroscience: neuronal spiking is often modeled as Brownian motion hitting a threshold, with the post-spike refractory period modeled as an exponentially decaying boundary, see \cite{Tamborrino2016}.

\section*{Known facts}

\begin{fact}[Existence and continuity of the density]
Since \(b(0) = b_0 > W_0\) and \(b\) is continuously differentiable on \((0,\infty)\), then according to \cite{Strassen1967, Durbin1985}, \(F\) admits a density \(f\) with respect to Lebesgue measure that is continuous on \((0,\infty)\).
\end{fact}

\section*{Open problem}

Despite the regularity theory above, no closed-form expression for \(f\) in
terms of elementary or standard special functions (Bessel, Airy, parabolic
cylinder, confluent hypergeometric, etc.) is known.

\begin{problem}
Find an explicit closed-form expression for the density
\(f(t)\), \(t > 0\), of the first-passage time
\[
  \tau = \inf\{\, t > 0 : W_t \ge b(t)\,\}
\]
of a standard Brownian motion to the exponentially decaying boundary
\(b(t) = b_0 e^{-ct}\).
\end{problem}

This stands in contrast to the handful of boundaries for which \(f\)
\emph{is} known explicitly; linear (Bachelier--L\'evy), square-root
\cite{Breiman1967}, and quadratic \cite{Salminen1988,Groeneboom1989}.

\section*{The Volterra integral equation formulation}

The natural rigorous substitute for an explicit formula is the
characterization of \(f\) as a solution of a Volterra integral equation of
the first kind, due to Peskir \cite{Peskir2002}.

\begin{fact}[Master integral equation]
Define \(H_{-1} = \phi\), the standard normal density, and
\(H_n(x) = \int_x^\infty H_{n-1}(z)\,dz\) for \(n \ge 0\). Then for every
integer \(n \ge -1\) and every \(t>0\),
\begin{equation}
  t^{n/2}\, H_n\!\left(\frac{b(t)}{\sqrt t}\right)
  \;=\;
  \int_0^t (t-s)^{n/2}\, H_n\!\left(\frac{b(t)-b(s)}{\sqrt{t-s}}\right) f(s)\, ds.
  \tag{$\ast$}
  \label{eq:master}
\end{equation}
This family holds for the exponential boundary \(b(t) = b_0 e^{-ct}\) for
every \(t > 0\), since \(b\) is continuously differentiable on all of
\((0,\infty)\) and Peskir's regularity hypotheses are satisfied for any such
boundary. (Here \(f(s)\,ds\) stands in for \(F(ds)\), valid because we have
already established that \(F\) admits a continuous density \(f\) for this
boundary; Peskir's original system holds in the more general
Lebesgue--Stieltjes form \(F(ds)\) for any continuous boundary, with no
density assumed --- see \cite{Peskir2002} for the general case.)
\end{fact}

The simplest member, \(n=-1\), with \(H_{-1}=\phi\), is the classical
\emph{generalized Abel equation} of Durbin \cite{Durbin1971}:
\begin{equation}
  \int_0^t \frac{1}{\sqrt{t-s}} \,
  \phi\!\left(\frac{b(t)-b(s)}{\sqrt{t-s}}\right) f(s)\, ds
  \;=\;
  \frac{1}{\sqrt{t}} \, \phi\!\left(\frac{b(t)}{\sqrt{t}}\right),
  \qquad \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}.
  \label{eq:durbin}
\end{equation}

For \(b(t) = b_0 e^{-ct}\), both \eqref{eq:master} and \eqref{eq:durbin}
are fully explicit in \(b(t) - b(s) = b_0\big(e^{-ct} - e^{-cs}\big)\) --- the
only unknown remaining is \(f\) inside the integral. The open problem above
is therefore equivalent to inverting \eqref{eq:master} in closed form for
this specific kernel.

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\bibitem{Tamborrino2016}
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\end{thebibliography}