An SDE of the form

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

is really short-hand notation for an equation involving Ito integrals:

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

Every book I have seen is stating that.

But that situation is unsettling to me. The parts of the SDE, i.e. $dX_t, \mu(X_t,t)dt$ and $\sigma(X_t,t)dB_t$ have no meaning on their one, although the notation suggests that they should have meaning. So

Is there a useful "differential" operator $d$ that can be applied to stochastic processes and gives meaning to the above SDE?

If the answer is in a sense negative, some sort of explanation on why that is would be appreciated.

  • $\begingroup$ I don’t think that anyone has developed a theory where there is a true meaning to them. The way they are used is just because it’s super convenient. But I’m not saying that you couldn’t do that, it just that at least I cant see immediately what good that would do. $\endgroup$ – Harto Saarinen Jul 15 '18 at 19:57
  • $\begingroup$ Also if you want to get rid of the $dB_t$ term you could ”cheat” a bit and replace it with $W_t dt$ where $W_t$ is a white noise. $\endgroup$ – Harto Saarinen Jul 15 '18 at 19:59

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