IsisCB Explore

Article ID: CBB077667833

On Itô’s Integral and the Fundamental Theorem of Stochastic Calculus (2023)

Begoña Fernández (Author)

Almagest
Volume: 14
Issue: 2
Pages: 136-148
Publication date: 2023
Language: English

DOI: 10.1484/J.ALMA.5.136762

URI: https://www.brepolsonline.net/doi/abs/10.1484/J.ALMA.5.136762

Outside Links

WorldCat

Google Scholar

JSTOR

In 1923, Wiener proved that the sample paths of Brownian Motion are almost surely nowhere differentiable and then, have infinite variation. This led to an interest in the study of continuous time stochastic processes. Initially, there were two main approaches: one based on Kolmogorov’s work on Markov processes, and another based on Levy’s approach to Brownian Motion. In the 1940’s, Kiyosi Ito began to investigate continuous time stochastic processes with independent increments. He found the way to define a Lebesgue-Stieltjes type integral, with respect to the Brownian Motion, and developed the now called Ito’s formula, thereby unifying both approaches and giving rise to the Theory of Stochastic Differential Equations. In this paper, we present the two different points of views, an integral with respect to the Brownian Motion developed previously by Wiener, and the construction of Ito’s Integral.

...More

Included in

Article Vincent Jullien; Carmen Martínez-Adame (2023) Foreward. Almagest (pp. 6-14). unapi

Citation URI

data.isiscb.org/p/isis/citation/CBB077667833

This citation is part of the Isis database.

Citations Indexes

Similar Citations

Article Siegmund-Schultze, Reinhard; (2007)
Philipp Frank, Richard von Mises, and the Frank-Mises (/p/isis/citation/CBB000780198/)

Article Ferraro, Giovanni; (2008)
The Integral as an Anti-Differential. An Aspect of Euler's Attempt to Transform the Calculus into an Algebraic Calculus (/p/isis/citation/CBB000933315/)

Chapter Archibald, Tom; (2003)
Differential equations: A historical overview to circa 1900 (/p/isis/citation/CBB000355275/)

Article Frédéric Jaëck; (2019)
Calcul différentiel et intégral adapté aux substitutions par Volterra (/p/isis/citation/CBB042003689/)

Article Marc Parmentier; (2015)
Leibniz et le virtuel (/p/isis/citation/CBB764317706/)

Article Cogliati, Alberto; (2014)
Early History of Infinite Continuous Groups, 1883--1898 (/p/isis/citation/CBB001420738/)

Chapter Guicciardini, Niccolò; (2003)
Newton's method and Leibniz's calculus (/p/isis/citation/CBB000355267/)

Article Olivier Darrigol; (2023)
A history of the relation between fluctuation and dissipation (/p/isis/citation/CBB193601353/)

Article Luca Peliti; Paolo Muratore-Ginanneschi; (2023)
R. Fürth’s 1933 paper “On certain relations between classical statistics and quantum mechanics” [“Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik”, Zeitschrift für Physik, 81 143–162] (/p/isis/citation/CBB934619261/)

Chapter Bigg, Charlotte; (2010)
A Visual History of Jean Perrin's Brownian Motion Curves (/p/isis/citation/CBB001221452/)

Article Bigg, Charlotte; (2008)
Evident Atoms: Visuality in Jean Perrin's Brownian Motion Research (/p/isis/citation/CBB000931140/)

Book George E. Smith; Raghav Seth; (2020)
Brownian Motion and Molecular Reality (/p/isis/citation/CBB800777900/)

Article Zambelli, Stefano; (2010)
Chemical Kinetics and Diffusion Approach: The History of the Klein--Kramers Equation (/p/isis/citation/CBB001022002/)

Article Nobukata Nagasawa; (2018)
On social and psychological aspects of a negligible reception of Natanson’s article of 1911 in the early history of quantum statistics (/p/isis/citation/CBB679249284/)

Article Berry, Dominic; (2015)
The Resisted Rise of Randomisation in Experimental Design: British Agricultural Science, c.1910--1930 (/p/isis/citation/CBB001510359/)

Article Vladimir Dragović; Irina Goryuchkina; (2020)
Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics (/p/isis/citation/CBB253445490/)

Article Fischer, Claude S.; (1987)
Technology's retreat: The decline of rural telephony in the United States, 1920-1940 (/p/isis/citation/CBB000040781/)

Article Sabrina Helena Bonfim; Sergio Roberto Nobre; (2020)
Historical Mathematical Study About Vector Calculus Introduction in Brazil: First Notes (/p/isis/citation/CBB564517999/)

Book Jan von Plato; (2017)
Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics (/p/isis/citation/CBB610005573/)

Article Luciano, Erika; (2006)
At the Origins of Functional Analysis: G. Peano and M. Gramegna on Ordinary Differential Equations (/p/isis/citation/CBB000770911/)