  In this paper we obtain a Wiener-Hopf type factorization for a real-valued
arithmetic Brownian motion with time-dependent drift and volatility. To the
best of our knowledge, this paper is the very first step towards realizing the
objective of deriving Wiener-Hopf type factorizations for (real-valued)
time-inhomogeneous Levy processes. In order to prove our main theorem, we
derive some new results regarding time-inhomogeneous noisy Wiener-Hopf
factorization. We demonstrate that in the special case of the arithmetic
Brownian motion with constant drift and volatility our main result agrees with
classical Wiener-Hopf factorization for this particular time-homogenous Levy
process.
