Pathwise Construction of Stochastic Integrals

TL;DR

This paper introduces a pathwise method to construct stochastic integrals that works across a wide range of probability measures without relying on measure dominance, applicable to any predictable integrand.

Contribution

It presents a novel measure-independent, pathwise construction of stochastic integrals that aligns with classical integrals under various probability measures.

Findings

Constructs stochastic integrals without a fixed probability measure.

Applies to any predictable integrand.

Ensures the constructed integral matches classical stochastic integrals.

Abstract

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.