It\^o Stochastic differentials

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.