Application of stochastic analysis in infinitely dimensional spaces in statistics?

Since stochastic analysis in infinitely dimensional spaces has been developed in the past decades, e.g. Hida distributions, Malliavin calculus, just to name a few. However, I have almost never seen that the models from forementioned fields have been treated from a statistician's perspective. To be precise, I mean here, as a statistician, his interest would be recovering the parameter of the model to which the observations are made, (knowns as inference, parameter identification, calibration, etc).

My questions can be summarized as follows.

1. Do there exist any statistical researches on inference of processes (or SDEs), using the techniques established in forementioned fields, namely different from (semi-)martingale approach? If so, could one suggest any references on the subject?

2. (side question) Why Malliavin calculus seems (much) more popular than Hida's distribution?