Pathwise stochastic integration with finite variation processes uniformly approximating c\`{a}dl\`{a}g processes

TL;DR

This paper introduces a new method for defining stochastic integrals using processes with finite variation that approximate cadlag paths, leading to a different correction term than classical integrals and including non-semimartingale examples.

Contribution

It constructs a novel class of finite variation processes for pathwise stochastic integration, extending classical results and providing new examples of non-semimartingale processes.

Findings

Defines a family of finite variation processes approximating cadlag paths.

Develops a stochastic integral with a unique correction term.

Provides examples of non-semimartingale processes via Skorohod maps.

Abstract

For any real-valued stochastic process $X$ with c\'rdl\'rg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-Zakai and Bichteler on pathwise stochastic integration. As a "byproduct" we obtain an example of a series of double Skorohod maps of a standard Brownian motion, which is not a semimartingale.