In *Stochastic Flows and Stochastic Differential Equations*, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in a Hölder space as long as the joint quadratic variation of $M(x,t)$ and $M(y,t)$ satisfies some regularity assumptions.

However, I don't understand his proof (regarding this, see my other question). Actually, I'm not even sure, if I understand his attempt.

I've searched the internet for a couple of weeks, but it seems like there is almost nothing on this topic. The few things I've found are papers and they just cite the statements of Kunita.

So, if anyone could explain his argument to me or come up with an other reference, I'd be highly thankful.

Thanks in advance to anybody who is trying to help.