Yes. A systematic study of stochastic (differential) algebra could be found in 

> Grenander, Ulf. [Probabilities on algebraic structures][1]. Courier Corporation, 2008.

Grenander studied the operation of integration on what are called "stochastic semi-groups". More specifically the Lie group representing the probability measures equipped with covariate derivatives(Lie derivative in most cases). If you want a geometric glimpse, you can have a look at [some reference geometric interpretation of general stochastic processes][2].

However, when you mentioned differential algebra, you are actually referring to a different object which is started by Kolchin et.al. [Differential Galois theory][3] is the correct name of the branch that studies the algebraic structure equipped with a derivation homomorphism.

The point here is that differential algebra does not provide too deep insight into the derivation homomorphism itself but focus on the D-module of derivation homomorphisms; however, the study of stochastic integration operators can be well addressed when we replaced the underlying measurable space and equipped with suitable Lie structure as shown by Grenander.


  [1]: https://books.google.gr/books?id=3eg5UzCivwoC&printsec=frontcover&hl=el&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
  [2]: http://mathoverflow.net/questions/264546/reference-for-feynman-kac/264562#264562
  [3]: http://mathoverflow.net/questions/201853/why-is-differential-galois-theory-not-widely-used