Itô Stochastic differentials

2020/04/07 by John A. Armstrong, Armstrong, John, Andrei C. Ionescu +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60H10 #FOS: Mathematics #Mathematical and Theoretical Analysis #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2004.03419

openalex publication_date 2020/04/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/01

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 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.

Related

• On the Functional Lévy-Itô Stochastic Calculus
• New stochastic calculus
• Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to Mathematical Finance
• The unique solution of stochastic differential equations
• Itô's formula, the stochastic exponential and change of measure on general time scales