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Generalization of Itô's formula for smooth nondegenerate martingales

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  • Moret, S.
  • Nualart, D.

Abstract

In this paper we prove the existence of the quadratic covariation [([not partial differential]F/[not partial differential]xk)(X), Xk] for all 1[less-than-or-equals, slant]k[less-than-or-equals, slant]d, where F belongs locally to the Sobolev space for some p>d and X is a d-dimensional smooth nondegenerate martingale adapted to a d-dimensional Brownian motion. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. As a consequence we obtain an extension of Itô's formula where the complementary term is one-half the sum of the quadratic covariations above.

Suggested Citation

  • Moret, S. & Nualart, D., 2001. "Generalization of Itô's formula for smooth nondegenerate martingales," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 115-149, January.
    • Handle: RePEc:eee:spapps:v:91:y:2001:i:1:p:115-149
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    References listed on IDEAS

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    1. Rozkosz, Andrzej, 1996. "Stochastic representation of diffusions corresponding to divergence form operators," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 11-33, October.
    2. Bardina, Xavier & Jolis, Maria, 1997. "An extension of Ito's formula for elliptic diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 83-109, July.
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    Cited by:

    1. Almada Monter, Sergio Angel, 2015. "Quadratic covariation estimates in non-smooth stochastic calculus," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 343-361.

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