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Itô stochastic differentials

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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. 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.
    2. John Armstrong, 2016. "The Markowitz Category," Papers 1611.07741, arXiv.org, revised Jun 2018.
    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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