The Annals of Probability

Central limit theorem for a Stratonovich integral with Malliavin calculus

Daniel Harnett and David Nualart

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Abstract

The purpose of this paper is to establish the convergence in law of the sequence of “midpoint” Riemann sums for a stochastic process of the form $f'(W)$, where $W$ is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an Itô integral of $f"(W)$ with respect to a Gaussian martingale independent of $W$. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for $q$-fold Skorohod integrals, which is a multi-dimensional extension of a result proved by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39–64]. The results proved in this paper are generalizations of previous work by Swanson [Ann. Probab. 35 (2007) 2122–2159] and Nourdin and Réveillac [Ann. Probab. 37 (2009) 2200–2230], who found a similar formula for two particular types of bifractional Brownian motion. We provide three examples of Gaussian processes $W$ that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters $H\le1/2$, $HK=1/4$. The others are Gaussian processes recently studied by Swanson [Probab. Theory Related Fields 138 (2007) 269–304], [Ann. Probab. 35 (2007) 2122–2159] in connection with the fluctuation of empirical quantiles of independent Brownian motion. In the first example the Gaussian martingale is a Brownian motion, and expressions are given for the other examples.

Article information

Source
Ann. Probab., Volume 41, Number 4 (2013), 2820-2879.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1372859768

Digital Object Identifier
doi:10.1214/12-AOP769

Mathematical Reviews number (MathSciNet)
MR3112933

Zentralblatt MATH identifier
1285.60050

Subjects
Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60F05: Central limit and other weak theorems 60G15: Gaussian processes

Keywords
Itô formula Skorohod integral Malliavin calculus fractional Brownian motion

Citation

Harnett, Daniel; Nualart, David. Central limit theorem for a Stratonovich integral with Malliavin calculus. Ann. Probab. 41 (2013), no. 4, 2820--2879. doi:10.1214/12-AOP769. https://projecteuclid.org/euclid.aop/1372859768


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References

  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [2] Burdzy, K. and Swanson, J. (2010). A change of variable formula with Itô correction term. Ann. Probab. 38 1817–1869.
  • [3] Houdré, C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. In Stochastic Models (Mexico City, 2002). Contemporary Mathematics 336 195–201. Amer. Math. Soc., Providence, RI.
  • [4] Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 619–624.
  • [5] Nourdin, I. and Nualart, D. (2010). Central limit theorems for multiple Skorokhod integrals. J. Theoret. Probab. 23 39–64.
  • [6] 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.
  • [7] Nourdin, I., Réveillac, A. and Swanson, J. (2010). The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter $1/6$. Electron. J. Probab. 15 2117–2162.
  • [8] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [9] Swanson, J. (2007). Weak convergence of the scaled median of independent Brownian motions. Probab. Theory Related Fields 138 269–304.
  • [10] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122–2159.
  • [11] Swanson, J. (2011). Fluctuations of the empirical quantiles of independent Brownian motions. Stochastic Process. Appl. 121 479–514.