Yes. A systematic study of stochastic (differential) algebra could be found in
Grenander, Ulf. Probabilities on algebraic structures. 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.
However, when you mentioned differential algebra, you are actually referring to a different object which is started by Kolchin et.al. Differential Galois theory 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.
