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Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Itô process $dX_t = \...

Let $(B^H_t)_{t\in [0,T]}$ be a fractional Brownian motion. We consider the following SDE where $b$ and $\sigma$ are Lipschitz $$X_t=x+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB^H_s.$$ When $H>1/2$, ...

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject. On page 14 of "A Course on Rough Paths With an ...

A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}_{s,t})=\frac{1}{2}...