Integration of branched rough paths

When the one-form is $Lip(\gamma -1)$ with $\gamma >p\ge 1$, we construct the integral of a branched $p$-rough path, which defines another branched $p$-rough path. We derive a quantitative bound for this integral and prove that it depends continuously on the driving branched rough path in rough path metric. Moreover, we prove that the first level branched rough integral coincides with a first level integral of the associated $\mathrm{\Pi }$-rough path.