Purely pathwise probability-free Ito integral

Abstract

This paper gives several simple constructions of the pathwise Ito integral $\int_0^t\phi d\omega$ for an integrand $\phi$ and a price path $\omega$ as integrator, with $\phi$ and $\omega$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $\phi$ nor $\omega$ are assumed to be paths of stochastic processes, and the Ito integral exists almost surely in a non-probabilistic financial sense. For example, one of the results shows the existence of $\int_0^t\phi d\omega$ for a cadlag integrand $\phi$ and a cadlag integrator $\omega$ with jumps bounded in a predictable manner.