  The solution of rough differential equation, driven by the It\^o signature of
a continuous local martingale, exists uniquely a.s. when the vector field is
Lip(\beta) for \beta > 1, and coincides a.s. with the It\^o signature of the
solution of parallel stochastic differential equation. Moreover, the It\^o
solution can be recovered pathwisely by concatenating discounted Stratonovich
solutions.
