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On Itô s formula for multidimensional Brownian motion
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  • Published: January 2000

On Itô s formula for multidimensional Brownian motion

  • Hans Föllmer1 &
  • Philip Protter2 

Probability Theory and Related Fields volume 116, pages 1–20 (2000)Cite this article

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  • 23 Citations

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Abstract.

Consider a d-dimensional Brownian motion X = (X 1,…,X d) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Itô s formula where the usual second order terms are replaced by the quadratic covariations [f k (X), X k] involving the weak first partial derivatives f k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X k] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, D-10099 Berlin, Germany. e-mail: foellmer@mathematik.hu-berlin.de, , , , , , DE

    Hans Föllmer

  2. Mathematics and Statistics Department, Purdue University, West Lafayette, IN 47907-1395, USA, , , , , , US

    Philip Protter

Authors
  1. Hans Föllmer
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  2. Philip Protter
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Received: 16 March 1998 / Revised version: 4 April 1999

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Föllmer, H., Protter, P. On Itô s formula for multidimensional Brownian motion. Probab Theory Relat Fields 116, 1–20 (2000). https://doi.org/10.1007/PL00008719

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  • Issue Date: January 2000

  • DOI: https://doi.org/10.1007/PL00008719

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  • Mathematics Subject Classification (1991): 60J65, 60H05, 60J45, 31C15, 31C25
  • Key words Itô's formula – Brownian motion – Stochastic integrals – Quadratic covariation – Dirichlet spaces – Polar sets
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