This note 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.  A basic version of It\^o's lemma shows that this definition agrees with F\"ollmer's. Finally, we  show the existence of \int_0^t\phi d\omega for a c\`adl\`ag  \phi with a finite variation index and a c\`adl\`ag integrator \omega with jumps bounded by 1 in absolute value .
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. A basic version of It\^o's lemma shows that our definition of the It\^o integral agrees with F\"ollmer's. Finally, we propose a tentative definition and show the existence of \int_0^t\phi d\omega for a c\`adl\`ag integrand \phi with a finite variation index and a c\`adl\`ag integrator \omega with jumps bounded in a predictable manner .
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note
paper
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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.
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this definition
our definition of the It\^o integral
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propose a tentative definition and
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integrand
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by 1 in absolute value
in a predictable manner