The Annals of Probability

A change of variable formula with Itô correction term

Krzysztof Burdzy and Jason Swanson

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Abstract

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F(t)=u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral g(F(t), t) dF(t) exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of F.

Article information

Source
Ann. Probab. Volume 38, Number 5 (2010), 1817-1869.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1282053773

Digital Object Identifier
doi:10.1214/09-AOP523

Mathematical Reviews number (MathSciNet)
MR2722787

Subjects
Primary: 60H05: Stochastic integrals
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Stochastic integration quartic variation quadratic variation stochastic partial differential equations long-range dependence iterated Brownian motion fractional Brownian motion self-similar processes

Citation

Burdzy, Krzysztof; Swanson, Jason. A change of variable formula with Itô correction term. Ann. Probab. 38 (2010), no. 5, 1817--1869. doi:10.1214/09-AOP523. https://projecteuclid.org/euclid.aop/1282053773.


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References

  • [1] Burdzy, K. and M̧adrecki, A. (1996). Itô formula for an asymptotically 4-stable process. Ann. Appl. Probab. 6 200–217.
  • [2] Cheridito, P. and Nualart, D. (2005). Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H∈(0, ½). Ann. Inst. H. Poincaré Probab. Statist. 41 1049–1081.
  • [3] Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24 698–742.
  • [4] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [5] Gradinaru, M., Nourdin, I., Russo, F. and Vallois, P. (2005). m-order integrals and generalized Itô’s formula: The case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 781–806.
  • [6] Gradinaru, M., Russo, F. and Vallois, P. (2003). Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index H≥¼. Ann. Probab. 31 1772–1820.
  • [7] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • [8] Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 619–624.
  • [9] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [10] Nourdin, I. and Réveillac, A. (2009). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4. Ann. Probab. 37 2200–2230.
  • [11] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [12] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.
  • [13] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
  • [14] Russo, F. and Vallois, P. (1995). The generalized covariation process and Itô formula. Stochastic Process. Appl. 59 81–104.
  • [15] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122–2159.