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Itô stochastic differentials

Author

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  • Armstrong, John
  • Ionescu, Andrei

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.

Suggested Citation

  • Armstrong, John & Ionescu, Andrei, 2024. "Itô stochastic differentials," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:spapps:v:171:y:2024:i:c:s0304414924000231
    DOI: 10.1016/j.spa.2024.104317
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    References listed on IDEAS

    as
    1. John Armstrong, 2016. "The Markowitz Category," Papers 1611.07741, arXiv.org, revised Jun 2018.
    2. Nualart, David & Saussereau, Bruno, 2009. "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 391-409, February.
    3. John Armstrong, 2018. "Classifying Financial Markets up to Isomorphism," Papers 1810.03546, arXiv.org, revised Jul 2020.
    Full references (including those not matched with items on IDEAS)

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