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An extension of Ito's formula for elliptic diffusion processes

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  • Bardina, Xavier
  • Jolis, Maria

Abstract

We prove an extension of Itô's formula for F(Xt, t), where F(x, t) has a locally square integrable derivative in x that satisfies a mild continuity condition in t, and X is a one-dimensional diffusion process such that the law of Xt has a density satisfying some properties. Following the ideas of Föllmer, et al. (1995), where they prove an analogous extension when X is the Brownian motion, the proof is based on the existence of a backward integral with respect to X. For this, conditions to ensure the reversibility of the diffusion property are needed. In a second part of this paper we show, using techniques of Malliavin calculus, that, under certain regularity on the coefficients, the extended Itô's formula can be applied to strongly elliptic and elliptic diffusions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:69:y:1997:i:1:p:83-109
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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.
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    Cited by:

    1. Ramin Okhrati & Uwe Schmock, 2015. "It\^o's formula for finite variation L\'evy processes: The case of non-smooth functions," Papers 1507.00294, arXiv.org.
    2. 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.

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