Notes on proofs of continuity theorem in rough path analysis

The solution Zz0,h of ODE driven by some smooth path h is a functional of h. Rough path analysis gives some new continuity theorem for the functional h → Zh. In the proof of [3], first, the solution is constructed on small interval and, iterating the procedure, the solution is constructed on the whole interval. In this procedure, the initial point of the solution in the second small interval is different from the first initial point and depends on h. However in the proof in [3], this seems not to be taken into the account. Also, the proof seems to work well in the case of bounded coefficients only. In this note, we consider the pairs of the solution itself and the derivative of the solution with respect to the initial point to prove the continuity theorem for the solution only itself. The coefficient of the ODE which the derivative satisfies is of linear growth. However, we can prove the continuity theorem for the pairs because the derivative of the solution is a solution of a certain linear ODE. Also, in application, it is necessary to consider the pairs of the solution itself and the derivative of the solution to study the H-derivative of Z. The aim of this note is to explain such a proof using the results in [3] in detail. The original version of this note is written in Japanese in April, 2005 as an additional reference for the participants of the probability summer school which was held in August, 2004.