# How to compare pathwise convergence and convergence in probability

This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here.

Motivation: It appears pathwise convergence can have a distinct advantage in the area of signal processing (I work on), specifically, we can improve filter operation https://en.wikipedia.org/wiki/Digital_filter. In fact I have written some matlab codes and applied it in real data, collected from sensor, and got some very interesting result while performing beamforming. But not sure how to prove the advantage of the algorithm mathematically. As I understand the pathwise convergence is possible through a graded vector spaces or direct sum of Bannach spaces. My algorithm is based on the functional on the element of individual Bannach spaces. I hope this makes sense.

EDIT: Motivation As I understand the difference between convergence in probability is more like global convergence and pathwise is like of local convergence. I am assuming that patwise convergence method gives some local infomation which is not there in the other methods which gives probability wise convergence. If this is true the local information must be embeded in the iterated integrals in case of rough path and in the model in regularity structure. I tried to extract these additional information (assuming it exists) from the model, which is also termed as signature in rough path literature.

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:The corresponidng equations showing probability wise convergence and pathwise convergence under the same setting would be highly informative. Would someone please help me.

Edit: The convergence referred specifically to PDE/SPDE. Really sorry if this question is annoying.

## 1 Answer

As I did not get any reply, I am trying to verify my understanding. This is related to convergence of SDE/PSDE through Ito type calculous and rough path or regularity structure. Any comments would be highly appreciated.

The main difference between "probability wise convergence" and "path wise convergence" is that the former achieves the convergence through " local calculations" and the other achieves the convergence through "global calculations". Both methods gives similar sort of convergence this means both method may give exact result for the same problem. However there is additional advantage through the path-wise method.

To elaborate the above an over-simplified example given below (but it might express the view). Consider we have stochastic process with $$n$$ points, each $$n$$ points is random variable. The "probability wise" convergence method would consider "characteristic" of each $$n$$ points. The "characteristic" is problem dependent and can be probability distribution of the variable at the points. On the other hand the path-wise convergence method would consider the characteristic of all $$n$$ points simultaneously. This is similar to the "model" in regularity structure and "signature" in the rough path theory.

Because point wise convergence can be achieved through the global model (iterated integral in case of rough path) these models carries important information which can be helpful.

Edit: An example that each level of the signature (model) carries some infomation can be deduced from the paper : https://arxiv.org/abs/1707.03993. In this paper it is shown how signature can be used as an input to RNN, a tool in machine learning area. Signature works as an input to the RNN as signature can be tought as features of a class. Now, the RNN will fail completely if we try to work only on 1st level of signature as there is not enouh informations on the first level. However, as we increase the number of level of signature into the RNN we expect to get better result, until we reach the point of overfitting.