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My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, namely the substitutions $$ (dt)^2 = (dB_t)(dt) = (dt)(dB_t) = 0 $$ $$ (dB_t)^2 = dt $$ and the stochastic product rule $$ d(X_tY_t) = (X_t)(dY_t) + (dX_t)(Y_t) + (dX_t)(dY_t) $$ Differential algebra deals with rings equipped with derivations, homomorphisms on the additive structure that obey the standard Liebnitz rule. From here there is a relatively strong differential Galois theory that describes how extension of differential fields behave when you adjoin solutions of ODEs.

My question is whether there has been any work on a stochastic differential algebra with the modified Liebnitz rule that can describe when solutions to SDEs might be beyond the scope of integrating/exponentiating rational functions/Brownian motion?

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  • $\begingroup$ In this formulation, you do a lot of confusions. In fact, It is the relations between stochastic integrals. This one is the It^o integral. $\endgroup$ – Zbigniew Feb 12 '17 at 13:07
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Yes. A systematic study of stochastic (differential) algebra could be found in

Grenander, Ulf. Probabilities on algebraic structures. Dover Books, 1981.

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. In fact, this idea is not new, the earliest motivation of studying the probability measures over an algebraic structure can be traced back to 1950s, the famous book

Parthasarathy, Kalyanapuram Rangachari. Probability measures on metric spaces. Vol. 352. American Mathematical Soc., 1967.

devoted two chapters extending the idea of studying the probability measures over locally compact groups, of which classic groups became natural candidates and hence the Lie derivative is yielded to describe the stochastic integrations. Actually this thinking is quite dominating in modern probability theory works, not only Pathasarathy's contribution but also later study carried out by Ledoux-Talagrand(they generalized the underlying space into Banch spaces instead of metric space) and Ambrosio (the study of gradient flow over the space consisting of probability measures.)

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.

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