Local solution method for the problem of enlargement of filtration

TL;DR

This paper introduces the local solution method for the enlargement of filtration problem, providing a unified approach to classical formulas in stochastic calculus, demonstrating its effectiveness and flexibility through multiple examples.

Contribution

The paper presents a new local solution method that unifies proofs of key formulas in the enlargement of filtration theory, enhancing understanding and application.

Findings

Unified proof of Jacod's formula

Effective technique demonstrated with examples

Method may require extensive computations

Abstract

The enlargement of filtration theory is a study of semimartingales when the basic filtration changes. This theory provides particular techniques on stochastic calculus. We present here a technique, that we call the local solution method. We will show, with several examples, that the local solution method is an effective and flexible method. In particular, with the local solution method, we will give a unified proof of three of the classical formulas, namely Jacod's formula, the progressive enlargement formula and the enlargement formula with honest time. We also show that, besides its generality, this method, necessiting long (but interesting) computations, may not be optimal.