Stochastic differential equations driven by a fractional brownian motion

Abstract

[en] This project is a general study of Stochastic Differential equations driven by a fractional Brownian motion of Hurst parameter $H>1 / 2$. Sections 3,4 and 5 follow the lines of [16] in order to define a stochastic integral with respect to the fractional Brownian motion and then, discussing the existence and uniqueness of solutions. The sixth section is a general discussion about Malliavin calculus with respect to the fractional Brownian motion that will be useful in sections 7 and 8 . Moreover, in section 6 we prove that by reinforcing the conditions on the coefficients, we obtain absolute continuity of the law of the solution in the same way as it is done in [14].

Section 7 is the application of the Malliavin calculus in order to bound the density function of the solution to a specific type of equations by using a general method constructed in [12]. Finally, section 8 is devoted to show all the work we weren't able to finish during the elaboration of this thesis. We decided to attack the problem of bounding the density of a general family of stochastic delay differential equations. The approach given in [12] turned out to be inefficient, so we decided to follow the same approach as in [1], [10] and [15].