Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper we consider solutions $\{X^H_t\}_{t\in \R_+}$ to stochastic differential equations driven by frac{t}ional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter~$H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon~$T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold. In both cases we exhibit the Lipschitz continuity w.r.t.~$H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to~$\tfrac{1}{2}$.