Girsanov theory under a finite entropy condition

TL;DR

This paper revisits Girsanov's theory using a simplified approach based on finite relative entropy, providing a self-contained presentation of standard results with improved mathematical tools.

Contribution

It introduces a new finite entropy-based method that simplifies the proof of Girsanov's theorem, replacing complex martingale representations with Riesz and Orlicz space techniques.

Findings

Standard Girsanov results are proved under finite entropy assumptions.

The approach simplifies proofs by using Riesz and Orlicz space representations.

The method applies to processes with and without jumps.

Abstract

This paper is about Girsanov's theory. It (almost) doesn't contain new results but it is based on a simplified new approach which takes advantage of the (weak) extra requirement that some relative entropy is finite. Under this assumption, we present and prove all the standard results pertaining to the absolute continuity of two continuous-time processes with or without jumps. We have tried to give as much as possible a self-contained presentation. The main advantage of the finite entropy strategy is that it allows us to replace martingale representation results by the simpler Riesz representations of the dual of a Hilbert space (in the continuous case) or of an Orlicz function space (in the jump case).