## Electronic Journal of Probability

### Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales

#### Abstract

Let $X$ be the fractional Brownian motion of any Hurst index $H\in (0,1)$ (resp. a semimartingale) and set $\alpha=H$ (resp. $\alpha=\frac{1}{2}$). If $Y$ is a continuous process and if $m$ is a positive integer, we study the existence of the limit, as $\varepsilon\rightarrow 0$, of the approximations $$I_{\varepsilon}(Y,X) :=\left\{\int_{0}^{t}Y_{s}\left(\frac{X_{s+\varepsilon}-X_{s}}{\varepsilon^{\alpha}}\right)^{m}ds,\,t\geq 0\right\}$$ of $m$-order integral of $Y$ with respect to $X$. For these two choices of $X$, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the $m$-th moment of the Gaussian standard random variable. In particular, if $m$ is an odd integer, the limit equals to zero. In this case, the convergence in distribution, as $\varepsilon\rightarrow 0$, of $\varepsilon^{-\frac{1}{2}} I_{\varepsilon}(1,X)$ is studied. We prove that the limit is a Brownian motion when $X$ is the fractional Brownian motion of index $H\in (0,\frac{1}{2}]$, and it is in term of a two dimensional standard Brownian motion when $X$ is a semimartingale.

#### Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 18, 26 p.

Dates
First available in Project Euclid: 23 May 2016

https://projecteuclid.org/euclid.ejp/1464037591

Digital Object Identifier
doi:10.1214/EJP.v8-166

Mathematical Reviews number (MathSciNet)
MR2041819

Zentralblatt MATH identifier
1063.60079

#### Citation

Gradinaru, Mihai; Nourdin, Ivan. Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales. Electron. J. Probab. 8 (2003), paper no. 18, 26 p. doi:10.1214/EJP.v8-166. https://projecteuclid.org/euclid.ejp/1464037591

#### References

• Alòs, E., Mazet, O., Nualart, D. (2000), Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $\frac{1}{2}$, Stoch. Proc. Appl., 86, 121-139.
• Alòs, E., León, J.A., Nualart, D. (2001) Stratonovich stochastic calculus for fractional Brownian motion with Hurst parameter lesser than $\frac{1}{2}$, Taiwanese J. Math, 5, 609-632.
• Cheredito, P., Nualart, D. (2003) Symmetric integration with respect to fractional Brownian motion Preprint Barcelona
• Coutin, L., Qian, Z. (2000) Stochastic differential equations for fractional Brownian motion, C. R. Acad. Sci. Paris, 330, Serie I, 1-6.
• Föllmer, H. (1981) Calcul d'Itô sans probabilités, Séminaire de Probabilités XV 1979/80, Lect. Notes in Math. 850, 143-150, Springer-Verlag.
• Giraitis, L., Surgailis, D., (1985) CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. verw. Gebiete, 70, 191-212.
• Gradinaru, M., Russo, F., Vallois, P., (2001) Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H<=1/4, To appear in Ann. Probab., 31.
• Gradinaru, M., Nourdin, I., Russo, F., Vallois, P. (2002) m-order integrals and Itô's formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index, Preprint IECN 2002-48
• Guyon, X., León, J. (1989) Convergence en loi des H-variations d'un processus gaussien stationnaire sur R, Ann. Inst. Henri Poincaré, 25, 265-282.
• Istas, J., Lang, G. (1997) Quadratic variations and estimation of the local H"older index of a Gaussian process, Ann. Inst. Henri Poincaré, 33, 407-436.
• Itô, K. (1951) Multiple Wiener integral, J. Math. Soc. Japan 3, 157-169.
• Jakubowski, A., Mémin, J., Pagès, G. (1989) Convergence en loi des suites d'intégrales stochastiques sur l'espace D1 de Skorokhod, Probab. Theory Related Fields, 81, 111-137.
• Nualart, D. (1995) The Malliavin calculus and related topics, Springer, Berlin Heidelber New-York.
• Nualart, D., Peccati, G. (2003) Convergence in law of multiple stochastic integrals, Preprint Barcelona.
• Revuz, D., Yor, M. (1994) Continuous martingales and Brownian motion, 2nd edition, Springer-Verlag.
• Rogers, L.C.G. (1997) Arbitrage with fractional Brownian motion, Math. Finance, 7, 95-105.
• Russo, F., Vallois, P. (1996) Itô formula for C1-functions of semimartingales, Probab. Theory Relat. Fields 104, 27-41.
• Taqqu, M.S. (1977) Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exibit a long range dependence, Z. Wahrsch. verw. Gebiete 40, 203-238.
• Taqqu, M.S. (1979) Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. verw. Gebiete 50, 53-83.
• Zähle, M. (1998) Integration with respect to fractal functions and stochastic calculus I., Probab. Theory Relat. Fields 111, 333-374.