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In this work we present different results concerning the signature and thecubature of fractional Brownian motion (fBm). The first result regards the rateof convergence of the expected signature of the linear piecewise approximationof the fBm to its exact value, for a value of the Hurst parameter$H\in(\frac{1}{2},1)$. We show that the rate of convergence is given by $2H$.We believe that this rate is sharp as it is consistent with the result of Niand Xu, who showed that the sharp rate of convergence for the Brownian motion(i.e. fBm with $H=\frac{1}{2}$) is given by $1$. The second result regards thebound of the coefficient of the rate of convergence obtained in the firstresult. We obtain an uniform bound for the coefficient for the $2k$-th term ofthe signature of $\frac{\tilde{A}k(2k-1)}{(k-1)!2^{k}}$, where $\tilde{A}$ is afinite constant independent of $k$. The third result regards the sharp decayrate of the expected signature of the fBm. We obtain a sharp bound for the$2k$-th term of the expected signature of $\frac{1}{k!2^{k}}$. The last resultsconcern the cubature method for the fBm for $H>\frac{1}{2}$. In particular, wedevelop the framework of the cubature method for fBm, provide a bound for theapproximation error in the general case, and obtain the cubature formula forthe fBm in a particular setting. These results extend the work of Lyons andVictoir, who focused on the Brownian motion case.

Some results on the Signature and Cubature of the Fractional Brownian motion for $H>\frac{1}{2}$