This thesis consists of two quite distinct topics. In the first and bigger part we show that the Mandelbrot-van Ness representation of fractional Brownian motion is almost surely smooth in the Hurst parameter H.
This dependence result is transferred to the solution of a stochastic differential equation driven by fractional Brownian motion if the stochastic differential equation is one-dimensional or H>1/2.
In the multidimensional case of H in (1/3, 1/2] we use rough path theory to make sense of the differential equations. However, despite it being possible to lift fractional Brownian motion as well as its derivative in H to a rough path via the limit of dyadic approximations, they cannot be lifted jointly in the same way. Nevertheless, we obtain that the solution to a rough stochastic differential equation driven by fractional Brownian motion is locally Lipschitz continuous in H.
In the last part of the thesis we define a directional Malliavin derivative connected to a continuous linear operator. We show that this directional Malliavin derivative being zero is equivalent to some measurability or independence condition on the random variable. Using this result, we obtain that two random variables, whose classical Malliavin derivatives live in orthogonal subspaces, are independent. We also extend the chain rule to directional Malliavin derivatives and a broader class of functions with weaker regularity assumptions.
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