Why are SDEs wrt different variables?

I am reading mathematical finance on my own. I am wondering how are differentiation wrt different variables defined.

In stochastic calculus, we can see equations like, $dX_{t} = \mu dt + \sigma dW_{t}$. But I am wondering how to differentiate some equations to get this. My questions arouse from my intuition from ODEs. In ODEs, an equation $y = f(x)$ only can be differentiated as, for example, $\frac{dy}{dx}=t$, then we can multiply $dx$ to both sides to get $dy = tdx$.

So what should $X_t$ look like, and especially the operator look like, in order for us to multiply it on both sides. In other words, given an arbitrary stochastic process, are we able to differentiate it?

Solution 1:

Consider the following stochastic process from time $[0, t]$ then given $dX_s = \mu ds + \sigma dW_s$ where $W_s$ is the standard Wiener process, we have

$\int_0^t dX_s = X_t - X_0 = \int_0^t \mu ds + \int_0^t \sigma dW_s$.

If the drift and diffusion coefficients are constants, we can make a further simplification to obtain $X_t = X_0 + \mu (t+0) + \sigma (W_t - W_0) = X_0 + \mu t + \sigma W_t$ because $W_0$ is defined to be $0$ almost surely.

There are differentiation rules for stochastic calculus, like the Ito's lemma. However, please note that we cannot differentiate stochastic processes (in the undergraduate calculus sense) that contain a Wiener process since the Wiener process is continuous everywhere but nowhere differentiable and consequential the differential of the stochastic process does not exist.