# Absolute Continuity of Semimartingales

## TL;DR

This paper establishes new equivalent conditions for the absolute continuity of laws of semimartingales, generalizing previous results by weakening assumptions and employing a generalized Girsanov's theorem, with applications to Itô-diffusions.

## Contribution

It introduces a generalized Girsanov's theorem and extends absolute continuity criteria to broader classes of semimartingales, including explosive cases.

## Key findings

- Provides a generalized Girsanov's theorem for semimartingales.
- Reproduces known results in one-dimensional Itô-diffusion cases.
- Offers a Khasminskii-type test for multi-dimensional Itô-diffusions.

## Abstract

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov's theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional It\^o-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multi-dimensional It\^o-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.