This question was deleted :https://math.stackexchange.com/questions/3478980/how-to-compare-pathwise-convergence-and-convergence-in-probability
Rough path theory which is a method to solve stochastic differential equation claims pathwise converse of the solution in contrast to Ito's integral and Stratonovich integral which claims convergence in probability. In addition regularity structure was introduced later to extend the method to solve stochastic partial differential equation. I understand that these methods works only under certain conditions. But my question is what are the differences we can obtain from a method which gives pathwise convergence compare to convergence in probability?
Edit: People are voting to close, without a comment, few comments would be appreciated.
Consider we are finding solution of the same problem by two different methods, A and B. Method A gives path-wise convergence and method B gives convergence in probability. I am trying to understand what can we infer from the solution of method A and solution of method B? Is there any different information we can infer from A compare to B? Can we somehow compare the solution (what we can learn) from A and B?
EDIT: Anyone who does not know mathematics but knows English, cannot understand what is unclear about this question. Can anyone tell me what is unclear. This is question about comparing 2 methods. Is these method does not exist? These methods cannot solve the same problem...etc what is it?