On It\^o differential equation in rough path theory

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.