## Stochastic calculus with respect to Gaussian processes.(English)Zbl 1015.60047

The authors consider a family of Gaussian processes $$(B_t)_{t\in {\mathbb R}_+}$$ of the form $$B_t = \int_0^t K(t,s) dW_s$$, where $$K$$ is a deterministic kernel and $$(W_t)_{t\in {\mathbb R}_+}$$ is a standard Wiener process. They construct a stochastic calculus with respect to such processes via the stochastic calculus of variations, using the anticipating Skorokhod integral operator with respect to $$(W_t)_{t\in {\mathbb R}_+}$$, which is denoted by $$\delta$$. The stochastic integral of an adapted process $$u$$ with respect to $$(B_t)_{t\in {\mathbb R}}$$ is defined to be $$\delta (K^* u)$$, where $$K^*$$ is the adjoint of the operator with kernel $$K$$. Itô and Stratonovich change of variable formulas and Hölder regularity results are proved for indefinite integrals with respect to $$(B_t)_{t\in {\mathbb R}}$$, for a wide class of deterministic (singular and regular) kernels $$K$$. The results apply in particular to fractional Brownian motion with Hurst parameter $$H\in (1/4,1/2)$$.

### MSC:

 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60G15 Gaussian processes
Full Text:

### 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. · Zbl 0914.60018 [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. · Zbl 1028.60047 [3] Carmona, P. and Coutin, L. (1998). Stochastic integration with respect to fractional Brownian motion. · Zbl 0921.60067 [4] Comte, F. and Renault, E. (1996). Long memory continuous time models. J. Econometrics 73 101-149. · Zbl 0856.62104 [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. · Zbl 0867.60029 [6] Decreusefond, L. and Üst ünel, A. S. (1998). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177-214. · Zbl 0924.60034 [7] Decreusefond, L. and Üst ünel, A. S. (1998). Fractional Brownian motion: theory and applications. ESAIM: Proceedings 5 75-86. · Zbl 0914.60019 [8] Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (1998). Stochastic calculus for fractional Brownian motion. I. Theory. · Zbl 0947.60061 [9] Feyel, D. and de la Pradelle, A. (1996). Fractional integrals and Brownian processes. Potential Anal. 10 273-288. · Zbl 0944.60045 [10] Gaveau, B. and Trauber, P. (1982). L’intégrale stochastic comme opérateur de divergence dans l’espace fonctionnel. J. Funct. Anal. 46 230-238. · Zbl 0488.60068 [11] Hu, Y. and Øksendal, B. (1999). Fractional white noise calculus and applications to finance. [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. · Zbl 0924.60042 [13] Lin, S. J. (1995). Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 121-140. · Zbl 0886.60076 [14] Malliavin, P. (1997). Stochastic Analysis. Springer, New York. · Zbl 0878.60001 [15] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422-437. JSTOR: · Zbl 0179.47801 [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. · Zbl 0955.60034 [17] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Probab. Appl. 21. · Zbl 0837.60050 [18] Nualart, D. (1998). Analysis on Wiener space and anticipating stochastic calculus. Lecture Notes in Math. 1690 123-227. Springer, New York. · Zbl 0915.60062 [19] Nualart, D. and Pardoux, E. (1988). Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 535-581. · Zbl 0629.60061 [20] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, New York. · Zbl 0818.26003 [21] Skorohod, A. V. (1975). On a generalization of a stochastic integral. Theory Probab. Appl. 20 219-233. · Zbl 0333.60060 [22] Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333-374. · Zbl 0918.60037 [23] Zähle, M. (1999). Integration with respect to fractal functions and stochastic calculus. II. · Zbl 0983.60054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.