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 $H>1/2$, the equation is in the Young's sence. and for $1/2<H<1/4$ is in rough path sence. My question: is the solution (i.e., $(X_t)_{t\in [0,T]}$) adapted to the natural filtration?
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1$\begingroup$ Yes, as is more or less obvious from uniqueness and the construction of solutions. Note that Lipschitz isn't enough for $\sigma$, you'll need at least $C^\alpha$ for $\alpha> 1/H$ to get unique solutions in general. $\endgroup$– Martin HairerFeb 9, 2021 at 13:48
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$\begingroup$ Thank you so much sir. It was helpful. $\endgroup$– yassine yassineFeb 9, 2021 at 15:58
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