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On Itô's formula for multidimensional Brownian motion


  • Föllmer, Hans
  • Protter, Philip E.


Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial derivatives fk of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), Xkj 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.

Suggested Citation

  • Föllmer, Hans & Protter, Philip E., 2001. "On Itô's formula for multidimensional Brownian motion," SFB 373 Discussion Papers 2001,90, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  • Handle: RePEc:zbw:sfb373:200190

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    References listed on IDEAS

    1. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    2. Gil-Alana, Luis A., 2000. "Mean reversion in the real exchange rates," Economics Letters, Elsevier, vol. 69(3), pages 285-288, December.
    3. Granger, C. W. J., 1981. "Some properties of time series data and their use in econometric model specification," Journal of Econometrics, Elsevier, vol. 16(1), pages 121-130, May.
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