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