Before revision: This paper gives a simple construction of the pathwise It\^o integral \int_0^t\phi d\omega for a continuous integrand \phi whose variation index is finite and a continuous price path \omega as integrator . The definition is pathwise in that neither \phi nor \omega are assumed to be paths of stochastic processes, and the It\^o integral exists almost surely in a non-probabilistic financial sense.
After revision: This paper gives a simple construction of the pathwise It\^o integral \int_0^t\phi d\omega for a continuous integrand \phi whose variation index is finite and a continuous 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 It\^o integral exists almost surely in a non-probabilistic financial sense.