In stochastic calculus, it is often standard to write a DE in differential form, such as $\mathrm dY = H \, \mathrm dX$ for the stochastic integral $$\displaystyle\int\limits_0^t H \, \mathrm d X := \displaystyle\int\limits_0^t H_s \, \mathrm d X_s.$$
The most common sense-interpretation of an expression like $\mathrm d A = \displaystyle\sum_{k=0}^{n-1} \alpha_{k, t} \, \mathrm dX_{k, t}$ I can think of is the stochastic integral expression
$$\displaystyle\sum_{k=0}^{n-1} \displaystyle\int\limits_0^t \alpha_{k, s} \, \mathrm d X_{k, s}.$$
For example: in common derivations of the Black–Scholes–Merton equation, one will see expressions like $$\mathrm d V=\left(\mu S_t \frac{\partial V}{\partial S_t}+\frac{\partial V}{\partial t}+\frac{ \sigma^2 S_t^2}{2} \frac{\partial^2 V}{\partial S_t^2}\right) \, \mathrm d t+\sigma S_t \frac{\partial V}{\partial S_t} \, \mathrm d W_t$$
appear. Thus, under this interpretation, $\mathrm d X_k$ is just the integrating function in the Itô integral.
Do differentials make sense in the stochastic case or is this just an abuse of notation?
In this answer, we can read that convenience is one major reason for the differential notation, but note that the respondent never assigns any meaning to stochastic differentials: "[I am] NEVER going to assign meaning to an expression like $\mathrm dW_t$ on its own. I will ONLY use it in a context where a corresponding integral expression makes sense."
Some literature in finance will define a stochastic differential as
$$\mathrm d z(t) := \lim _{\Delta t \to 0} \Delta z=\lim _{\Delta t \to 0} \sqrt{\Delta t} \tilde{\epsilon},$$
but clearly this is the same kind of informal kludgely hand-waving (the limit is obviously zero!) you hear about before learning about how differentials in analysis are constructed (via the exterior derivative, $k$-forms, etc.).
Some books even include expressions involving sqaures of differentials,
$$(\mathrm dB_t)^2 = (B_{t+\mathrm dt}-B_t)^2 = \mathrm dt$$
which are also left vacantly undefined.
One can first attempt to formalize random differentials in the following manner: assume a smooth manifold $\mathcal M$. Now consider a fiber bundle $\pi \colon E \to \mathcal M$, in this case we want the cotangent space $E \cong T^*\mathcal M$.
We can then define the randomization of $\Gamma(E) \cong \operatorname{Hom}\left(\mathcal M, E\right)$ as $\Gamma(E) \mapsto \Omega \times \Gamma(E)$, for some probability space $\Omega$. Finally a random $1$-form is just a map
$$\begin{align*}\Omega \times M &\to \Gamma(T^* \mathcal M) \cong \operatorname{Hom}\left(\mathcal M, T^*\mathcal M\right) \\ \gamma &\colon (\omega, x) \mapsto \alpha_\omega \end{align*},$$
giving us one form to integrate for every $\omega \in \Omega$. The map $\gamma$ could then be integrated against, yielding a random variable:
$$\displaystyle\int_{\mathcal M} \gamma(\omega, x) = \mathbb X(\omega).$$
Now, this only yields a random variable $\mathbb X$ and not a stochastic process $\left\{X_t \right\}_t$.
As the theory the theory of forms is generally very well-understood, it seems like the generalization to stochastic forms should not be that hard. The machinery and tools differential forms generalize to the language of chain complexes, co-chain complexes, cohomology, the cup product $\smile$ and cap product $\frown$, the $\operatorname{Tor}$ and $\operatorname{Ext}$ functor, and so on. It seems to be very useful to be able to be able to express your theory in these terms, hence my motivation for trying to formalize stochastic differentials.
Can we formalize stochastic differentials, have this been done and what constructions have been attempted? How can we understand a symbol like $\mathrm dW_t$?
[2] Hsu, E. P. (2002). Stochastic analysis on manifolds. American Mathematical Society.
– Alex Apr 16 '24 at 08:13