Bernoulli

  • Bernoulli
  • Volume 13, Number 3 (2007), 820-830.

On Itô’s formula for elliptic diffusion processes

Xavier Bardina and Carles Rovira

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Abstract

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83–109] 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 certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303–328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function F has a locally integrable derivative in t, we can avoid the mild continuity condition in t for the derivative of F in x.

Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 820-830.

Dates
First available in Project Euclid: 7 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186503488

Digital Object Identifier
doi:10.3150/07-BEJ6049

Mathematical Reviews number (MathSciNet)
MR2348752

Zentralblatt MATH identifier
1133.60024

Keywords
diffusion processes integration with respect to local time Itô’s formula local time

Citation

Bardina, Xavier; Rovira, Carles. On Itô’s formula for elliptic diffusion processes. Bernoulli 13 (2007), no. 3, 820--830. doi:10.3150/07-BEJ6049. https://projecteuclid.org/euclid.bj/1186503488


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