The Ito Formula

  • Gopinath Kallianpur
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 13)


A process M t = (M t 1, ... ,M t d ) taking values in R d is a martingale with respect to the increasing σ-field family (F t ) if (M t i , F t ) is a martingale for each i = 1, ... ,d, or equivalently, if (θ,M t ,) is a real-valued martingale with respect to (F t ) for every θ, ∈ R d . Here we use (,) to denote inner product in R d . (M t , F t ) with M 0 = 0 (a.s.) is a d-dimensional, continuous L 2-martingale if for every θR d , (θ,M t ) is a continuous L 2-martingale with respect toF t . It is then easy to verify the existence of a unique d × d-matrix—valued process A t , = (A t ij ) with the following properties:
  1. a.

    Each A t ij is F t -measurable.

  2. b.

    A 0 = 0 and A t (ω) is continuous in t for almost all ω.

  3. c.

    For θR d , (A t θ,θ) is the (continuous) increasing process associated with (θ,M t ).



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  1. The proof of Theorem 4.2.1 is due to Meyer [44]. The variant of Ito’s formula given in Theorem 4.2.2 is due to Neveu [73].Google Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Gopinath Kallianpur
    • 1
  1. 1.Department of StatisticsUniversity of North CarolinaChapel HillUSA

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