On pathwise uniform approximation of processes with c\`adl\`ag trajectories by processes with finite total variation

TL;DR

This paper introduces a method to approximate cadlag stochastic processes with finite total variation processes, enabling pathwise integration and decomposition of processes with infinite variation.

Contribution

It defines a family of finite variation processes that approximate cadlag processes, facilitating new stochastic integral constructions and process decompositions.

Findings

Decomposition of cadlag processes into finite variation and small amplitude infinite variation parts.

Definition of stochastic integral as a limit of pathwise Lebesgue-Stieltjes integrals.

New approach to process approximation and stochastic integration with correction terms.

Abstract

For any real-valued stochastic process X with c\`adl\`ag paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with c\`adl\`ag paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term.