A Girsanov Result through Birkhoff Integral

TL;DR

This paper extends the Girsanov theorem to vector-valued processes using Birkhoff integral, providing a measure-theoretic framework and a martingale representation for the modified process.

Contribution

It introduces a vector-valued Girsanov theorem leveraging Birkhoff integral and establishes a martingale representation for the transformed process.

Findings

Established a vector-valued Girsanov theorem.

Developed a measure-theoretic approach using Birkhoff integral.

Constructed a martingale equivalent for the transformed process.

Abstract

A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one.