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.
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.
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a simple construction
several simple constructions
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It\^o
Ito
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a continuous integrand \phi whose variation index is finite and a continuous
an integrand \phi and a
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. The definition is
, with \phi and \omega satisfying various topological and analytical conditions. The definitions are purely
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It\^o
Ito
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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
For example, one of the results shows the
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c\`adl\`ag integrand \phi with a finite variation index and a c\`adl\`ag
cadlag integrand \phi and a cadlag