Itô Stochastic Differentials

John Armstrong, Andrei Ionescu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

We give an infinitesimal meaning to the symbol dXt 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 Itô integration theory. In particular, we link our notion of a differential with Itô 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 Itô’s lemma. We apply our results to financial mathematics to give a theory of portfolios at an instant in time.
Original languageEnglish
JournalStochastic Processes and Their Applications
Publication statusAccepted/In press - 8 Feb 2024

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