Existence and upper bound for the density of solutions of stochastic differential equations driven by generalized grey noise

Abstract

In this paper we establish the absolute continuity of the law of the solution of stochastic differential equations driven by generalised grey noise under the Hörmander condition. We then derive an upper bound and show the smoothness of the density. Finally we study the stability problem of the solutions.

Keywords:

• Generalised grey Brownian motion
• fractional Brownian motion
• stochastic differential equations
• absolute continuity

Acknowledgements

We would like to thank Professor David Nualart for reading the first version of this paper and the suggestion to study a more general case presented here.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was Financially supported by the project UID/MAT/04674/2013 and Laboratory LIBMA form the University Cadi Ayyad Marrakech are gratefully acknowledged. Finally, we thank Habib Ouerdiane for constant support and organization of the Hammamet conference.