## The Annals of Probability

### Stochastic Calculus with Respect to Gaussian Processes

#### Abstract

In this paper we develop a stochastic calculus with respect to a Gaussian process of the form $B_t = \int^t_0 K(t, s)\, dW_s$, where $W$ is a Wiener process and $K(t, s)$ is a square integrable kernel, using the techniques of the stochastic calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums.The particular case of the fractional Brownian motion is discussed.

#### Article information

Source
Ann. Probab. Volume 29, Number 2 (2001), 766-801.

Dates
First available: 21 December 2001

http://projecteuclid.org/euclid.aop/1008956692

Digital Object Identifier
doi:10.1214/aop/1008956692

Mathematical Reviews number (MathSciNet)
MR1849177

Zentralblatt MATH identifier
1015.60047

#### Citation

Alòs, Elisa ,1 2; and Mazet, Olivier; Nualart, David. Stochastic Calculus with Respect to Gaussian Processes. The Annals of Probability 29 (2001), no. 2, 766--801. doi:10.1214/aop/1008956692. http://projecteuclid.org/euclid.aop/1008956692.

#### References

• [1] Al os, E. and Nualart, D. (1998). An extension of It o's formula for anticipating processes. J. Theoret. Probab. 11 493-514.
• [2] Al os, E., Mazet, O. and Nualart, D. (2000). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 12. Stochastic Process. Appl. 86 121-139.
• [3] Carmona, P. and Coutin, L. (1998). Stochastic integration with respect to fractional Brownian motion. Preprint.
• [4] Comte, F. and Renault, E. (1996). Long memory continuous time models. J. Econometrics 73 101-149.
• [5] Dai, W. and Heyde, C. C. (1996). It o's formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal. 9 439-448.
• [6] Decreusefond, L. and ¨Ust ¨unel, A. S. (1998). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177-214.
• [7] Decreusefond, L. and ¨Ust ¨unel, A. S. (1998). Fractional Brownian motion: theory and applications. ESAIM: Proceedings 5 75-86.
• [8] Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (1998). Stochastic calculus for fractional Brownian motion. I. Theory. Preprint.
• [9] Feyel, D. and de la Pradelle, A. (1996). Fractional integrals and Brownian processes. Potential Anal. 10 273-288.
• [10] Gaveau, B. and Trauber, P. (1982). L'int´egrale stochastic comme op´erateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46 230-238.
• [11] Hu, Y. and Øksendal, B. (1999). Fractional white noise calculus and applications to finance. Preprint.
• [12] Kleptsyna, M. L., Kloeden, P. E. and Anh, V. V. (1998). Existence and uniqueness theorems for stochastic differential equations with fractal Brownian motion. Problemy Peredachi Informatsii 34 54-56.
• [13] Lin, S. J. (1995). Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 121-140.
• [14] Malliavin, P. (1997). Stochastic Analysis. Springer, New York.
• [15] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422-437.
• [16] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsano formula and other analytical results on fractional Brownian motion. Bernoulli 5 571-587.
• [17] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Probab. Appl. 21.
• [18] Nualart, D. (1998). Analysis on Wiener space and anticipating stochastic calculus. Lecture Notes in Math. 1690 123-227. Springer, New York.
• [19] Nualart, D. and Pardoux, E. (1988). Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 535-581.
• [20] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, New York.
• [21] Skorohod, A. V. (1975). On a generalization of a stochastic integral. Theory Probab. Appl. 20 219-233.
• [22] Z¨ahle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333-374.
• [23] Z¨ahle, M. (1999). Integration with respect to fractal functions and stochastic calculus. II. Preprint.