The Martingale Representation Theorem and Clark-Ocone formula

TL;DR

This paper investigates the Martingale Representation Theorem and Clark-Ocone formula, extending their applicability beyond regular Sobolev spaces and exploring their use with various stochastic processes.

Contribution

It extends the classical theorems to broader contexts involving measure changes and filtration enlargements, covering Brownian, jump, and Levy process-driven martingales.

Findings

Extended theorems beyond Sobolev spaces

Applied results to jump and Levy processes

Analyzed measure changes and filtration enlargements

Abstract

In this paper we explore the fundamentals of the Martingale Representation Theorem (MRT) and a closely related result, the Clark-Ocone formula. We also investigate how far these theorems can be taken, notably beyond the regular Sobolev spaces, through changes of measures and enlargement of filtrations. We look at Brownian motion (B.M.) driven continuous martingales as well as Jump and Levy process-driven martingales.