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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 $\mathrm dW_t$?

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    $\begingroup$ The first sentence in your post answers the question. The answer to the last sentence is: why should we? Edit: just a few weeks ago someone here on MSE claimed that $dt$ and $dW_t$ are differential forms. When I asked back what the wedge product of the two is the post got deleted by the author. $\endgroup$
    – Kurt G.
    Apr 15 at 18:27
  • $\begingroup$ One can actually make sense of equations like $(dB)^2=dt$ without integrals - $d$ stands for a finite (non-infinitesimal) difference and the equality is modulo a suitable ideal (something like $o(dt)$). See the book Quantum Fluctuations by Edward Nelson (there might be better references but this is the one that I know). His book Radically Elementary Probability Theory has a similar (though less standard) point of view. [I am definitely not an expert, but you might find it helpful.] $\endgroup$
    – user8268
    Apr 15 at 18:48
  • $\begingroup$ More of a general comment: Ikeda and Watanabe's book explicitly treat the algebraic structure of Ito and Stratonovich differentials $dX$. They form a ring IIRC. For the most part to prove any of the standard rules, e.g. $(dB)^2 = dt$ or $(X+Y)dZ=...$, you need to resort back to the integrals, but once that is done you can absolutely just work in the space of differentials which is what many applied practitioners do. $\endgroup$ Apr 15 at 19:02
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    $\begingroup$ I will donate a bottle of absinthe to the first person who explains to me what the two-form $dt\wedge dW_t$ is. $\endgroup$
    – Kurt G.
    Apr 15 at 19:13
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    $\begingroup$ I'll match the donation by @KurtG. with an extra bottle of absinthe $\endgroup$ Apr 15 at 19:19

1 Answer 1

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Differentials in stochastic calculus have very precise interpretations. The stochastic differential equation in differential form

$$dX_t = \mu (t, X_t) dt + \sigma (t, X_t) dB_t$$

translates precisely to its integral form:

$$X_t - X_0 = \int_0^t \mu (s, X_s) ds + \int_0^t \sigma (s, X_s) dB_s$$

The box calculus rules $(dB_t)^2 = dt$ and $dB_t dt = dtdB_t = (dt)^2 = 0$ are short forms for quadratic variation for Brownian and covariation of Brownian motion and a path of bounded variation, respectively.

Any reasonable book I've encountered on stochastic calculus explicitly defines this convention; I have never seen it left "vacantly undefined". In fact, the book you cite makes this formalism precise immediately after introducing it.

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    $\begingroup$ I can confirm this answer. My advisor, who is a probablist, taught me when you see a SDE in differential form, it really stands for an integral equation. $\endgroup$ Apr 16 at 3:09
  • $\begingroup$ Apart from being this shotthand of notation, do differentials carry a meaning of their own? Does $\mathrm dW_t$ have any interpretation as a symbol itself? I mentioned the common interpretation as an abuse of notation (shorthand for a stochastic integral) in the question. $\endgroup$ Apr 16 at 14:45
  • $\begingroup$ @MarkusKlyver I wouldn't call it an abuse of notation, because it isn't. It's rather a very useful shorthand. I can't think of a sensible meaning to ascribe to it akin to a differential form. One can exhibit stochastic integration as integration against white noise, where "$dW_t / dt$" is given a meaning in the distributional sense. $\endgroup$ Apr 17 at 15:41

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