Aspects of Stochastic Integration with Respect to Processes of Unbounded p-variation

TL;DR

This paper establishes the existence of stochastic integrals involving functions with discontinuities and unbounded p-variation, extending classical results by Young through pathwise methods under certain regularity conditions.

Contribution

It introduces conditions under which stochastic integrals with discontinuous integrands of unbounded p-variation can be defined pathwise, surpassing limitations of Young and rough path theories.

Findings

Proves existence of integrals for discontinuous functions with unbounded p-variation.

Extends Young's classical results to broader classes of stochastic integrals.

Provides pathwise construction under regularity assumptions on processes.

Abstract

This paper deals with stochastic integrals of form $\int_0^T f(X_u)d Y_u$ in a case where the function $f$ has discontinuities, and hence the process $f(X)$ is usually of unbounded $p$-variation for every $p\geq 1$. Consequently, integration theory introduced by Young or rough path theory introduced by Lyons cannot be applied directly. In this paper we prove the existence of such integrals in a pathwise sense provided that $X$ and $Y$ have suitably regular paths together with some minor additional assumptions. In many cases of interest, our results extend the celebrated results by Young.