It\^o Stochastic differentials

John Armstrong; Andrei Ionescu

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arXiv · 2004.03419

It\^o Stochastic differentials

John Armstrong·Andrei Ionescu

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Abstract

We give an infinitesimal meaning to the symbol $dX_t$ for a continuous semimartingale $X$ at an instant in time $t$. We define a vector space structure on the space of differentials at time $t$ and deduce key properties consistent with the classical It\^o integration theory. In particular, we link our notion of a differential with It\^o integration via a stochastic version of the Fundamental Theorem of Calculus. Our differentials obey a version of the chain rule, which is a local version of It\^o's lemma. We apply our results to financial mathematics to give a theory of portfolios at an instant in time.

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John Armstrong, Andrei Ionescu. 2020-04-07. It\^o Stochastic differentials. https://arxiv.org/abs/2004.03419

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