Let $S_t$ be a continuos semimartingale, that is we have a decomposition $$S_t = A_t + M_t + C,$$
where $A$ is of finite variation and $M_t$ is a local martingale.
Ito calculus allows us to make sense of the stochastic integral $ \int_0^T X_s dS_s$ in this situation. Now, when one considers the Langevin-type stochastic differential equation
$$ dX_t = f(X_t) dt + d S_t$$
this means that we have
$$ X_t = X_0 \int_0^t f(X_s) ds + \int_0^T dS_s = X_0 + \int_0^t f(X_s) ds + S_t.$$
I was wondering if one can in any sense get a higher-order differential equation from this. If $S_s$ was differentiable, then one would get $$ \overset{\cdot}{X_t} = f(X_t) + \frac{d}{dt} S_t$$
which would be equialent to the new SDE
$$ d \overset{\cdot}{X_t} = d f(X_t) + d \frac{d}{dt} S_t.$$
However, it is a common exercise to show that Brownian motion is not differentiable, which excludes the most common cases. Of course, if one chooses the local martingale to be zero ($M_t=0$), then one can just choose $A_t$ to be differentiable and this would work, but then essentially obtain an ODE. So, my question would be if there is anything in between, i.e. a semimartingale which has some local martingale contribution and is differentiable?
And if this does not exist, is there any other way to make sense of the differentiation of the original SDE to obtain equations of higher order?