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

TL;DR

This paper develops a Wiener-Hopf type factorization for arithmetic Brownian motion with time-dependent parameters, advancing the understanding of time-inhomogeneous Levy processes and extending classical results.

Contribution

It introduces the first Wiener-Hopf factorization for real-valued time-inhomogeneous Levy processes, specifically for arithmetic Brownian motion with variable drift and volatility.

Findings

Derived new results on time-inhomogeneous noisy Wiener-Hopf factorization.

Showed that the new factorization reduces to classical results in the time-homogeneous case.

Established a foundational step towards Wiener-Hopf factorizations for broader classes of processes.

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.