Bernoulli

  • Bernoulli
  • Volume 13, Number 3 (2007), 695-711.

Correcting Newton–Côtes integrals by Lévy areas

Ivan Nourdin and Thomas Simon

Full-text: Open access

Abstract

In this note we introduce the notion of Newton–Côtes functionals corrected by Lévy areas, which enables us to consider integrals of the type f(y) dx, where f is a C2m function and x, y are real Hölderian functions with index α>1/(2m+1) for all m∈ℕ*. We show that this concept extends the Newton–Côtes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by x, interpreted using the symmetric Russo–Vallois integral.

Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 695-711.

Dates
First available in Project Euclid: 7 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186503483

Digital Object Identifier
doi:10.3150/07-BEJ6015

Mathematical Reviews number (MathSciNet)
MR2348747

Zentralblatt MATH identifier
1132.60047

Keywords
fractional Brownian motion Lévy area Newton–Côtes integral rough differential equation symmetric stochastic integral

Citation

Nourdin, Ivan; Simon, Thomas. Correcting Newton–Côtes integrals by Lévy areas. Bernoulli 13 (2007), no. 3, 695--711. doi:10.3150/07-BEJ6015. https://projecteuclid.org/euclid.bj/1186503483


Export citation

References

  • Alòs, E. and Nualart, D. (2002). Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 129–152.
  • Cheridito, P. (2003). Arbitrage in fractional Brownian motion models. Finance and Stochastics 7 533–553.
  • Cheridito, P. and Nualart, D. (2005). Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter $H \in(0,1/2)$. Ann. Inst. H. Poincaré Probab. Statist. 41 1049–1081.
  • Comte, F. and Renault, E. (1998). Long memory in continuous time volatility models. Math. Finance 8 291–323.
  • Coutin, L. (2007). An introduction to (stochastic) calculus with respect to fractional Brownian motion. Séminaire de Probabilités XL. To appear.
  • Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140.
  • Cutland, N., Kopp, P. and Willinger, W. (1995). Stock price returns and the Joseph effect: a fractional version of the Black–Schole model. In Seminar on Stochastic Analysis, Random Fields and Applications, Progr. Probab. 36 327–351.
  • Errami, M. and Russo, F. (2003). $n$-Covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 259–299.
  • Feyel, D. and De la Pradelle, A. (2006). Curvilinear integral along enriched paths. Electron. J. Probab. 11 860–892.
  • Gradinaru, M., Nourdin, I., Russo, F. and Vallois, P. (2005). $m$-Order integrals and Itô's formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 781–806.
  • Lyons, T.J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215–310.
  • Lyons, T.J. and Qian, Z. (2003). System Controls and Rough Paths. Oxford: Oxford University Press.
  • Neuenkirch, A. and Nourdin, I. (2007). Exact rate of convergence of some approximation schemes associated with SDEs driven by a fBm. J. Theoret. Probab. To appear.
  • Nourdin, I. (2007). A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. Séminaire de Probabilités XLI. To appear.
  • Nourdin, I. and Simon, T. (2006). On the absolute continuity of one-dimensional SDE's driven by a fractional Brownian motion. Statist. Probab. Letters 76 907–912.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. New York: Springer.
  • Nualart, D. (2003). Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 3–39.
  • Nualart, D. and Rasçanu, A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53 55–81.
  • Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.