Wiener-Hopf Factorization for Arithmetic Brownian Motion with Time-Dependent Drift and Volatility

Abstract

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.