Proving existence results in martingale theory using a subsequence principle

TL;DR

This paper introduces new proofs for key existence results in martingale theory using a subsequence principle, simplifying the construction of stochastic integrals and advancing theoretical understanding.

Contribution

It provides novel proofs for the existence of the compensator and quadratic variation in martingale theory via a functional analytic subsequence approach.

Findings

Simplified proofs of the existence of the compensator and quadratic variation.

Application of subsequence principle to stochastic integral construction.

Enhanced theoretical framework for martingale and semimartingale analysis.

Abstract

New proofs are given of the existence of the compensator (or dual predictable projection) of a locally integrable c\'adl\'ag adapted process of finite variation and of the existence of the quadratic variation process for a c\'adl\'ag local martingale. Both proofs apply a functional analytic subsequence principle. After presenting the proofs, we discuss their application in giving a simplified account of the construction of the stochastic integral of a locally bounded predictable process with respect to a semimartingale.