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

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 $p\ge 1$. 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.