## Electronic Journal of Probability

### Skorohod and Stratonovich integration in the plane

#### Abstract

This article gives an account on various aspects of stochastic calculus in the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of variable formula for a path $x:[0,1]^{2}\to\mathbb{R}$ satisfying some Hölder regularity conditions with a Hölder exponent greater than $1/3$. (ii) Get some Skorohod change of variable formulas for a large class of Gaussian processes defined on $[0,1]^{2}$. (iii) Compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the respective change of variable formulas.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 39, 39 pp.

Dates
Accepted: 8 April 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067145

Digital Object Identifier
doi:10.1214/EJP.v20-3041

Mathematical Reviews number (MathSciNet)
MR3335830

Zentralblatt MATH identifier
1322.60081

Rights

#### Citation

Chouk, Khalil; Tindel, Samy. Skorohod and Stratonovich integration in the plane. Electron. J. Probab. 20 (2015), paper no. 39, 39 pp. doi:10.1214/EJP.v20-3041. https://projecteuclid.org/euclid.ejp/1465067145

#### References

• Alos, Elisa; Mazet, Olivier; Nualart, David. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), no. 2, 766–801.
• Bardina, Xavier; Jolis, Maria. Multiple fractional integral with Hurst parameter less than $\frac 12$. Stochastic Process. Appl. 116 (2006), no. 3, 463–479.
• Cairoli, R.; Walsh, John B. Stochastic integrals in the plane. Acta Math. 134 (1975), 111–183.
• K.Chouk, M. Gubinelli: Rough sheets. Work in progress (2013).
• Da Prato, Giuseppe; Malliavin, Paul; Nualart, David. Compact families of Wiener functionals. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 12, 1287–1291.
• Friz, Peter; Riedel, Sebastian. Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 1, 154–194.
• Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths. Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0
• Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86–140.
• Gubinelli, Massimiliano; Tindel, Samy. Rough evolution equations. Ann. Probab. 38 (2010), no. 1, 1–75.
• Hu, Yaozhong; Jolis, Maria; Tindel, Samy. On Stratonovich and Skorohod stochastic calculus for Gaussian processes. Ann. Probab. 41 (2013), no. 3A, 1656–1693.
• Hu, Yao-zhong; Yan, Jia-an. Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 399–414.
• Leon, Jorge A.; Nualart, David. An extension of the divergence operator for Gaussian processes. Stochastic Process. Appl. 115 (2005), no. 3, 481–492.
• Nualart, David. Une formule d'Itô pour les martingales continues à deux indices et quelques applications. (French) [An Ito formula for continuous two-parameter martingales and some applications] Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 3, 251–275.
• Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
• Quer-Sardanyons, Lluis; Tindel, Samy. The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stochastic Process. Appl. 117 (2007), no. 10, 1448–1472.
• Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
• Tudor, Ciprian A.; Viens, Frederi G. Itô formula and local time for the fractional Brownian sheet. Electron. J. Probab. 8 (2003), no. 14, 31 pp. (electronic).
• Tudor, Ciprian A.; Viens, Frederi G. Itô formula for the two-parameter fractional Brownian motion using the extended divergence operator. Stochastics 78 (2006), no. 6, 443–462.
• Wong, Eugene; Zakai, Moshe. Differentiation formulas for stochastic integrals in the plane. Stochastic Processes Appl. 6 (1977/78), no. 3, 339–349.