  In this paper we prove that every random variable of the form $F(M_T)$ with
$F:\real^d \to\real$ a Borelian map and $M$ a $d$-dimensional continuous Markov
martingale with respect to a Markov filtration $\mathcal{F}$ admits an exact
integral representation with respect to $M$, that is, without any orthogonal
component. This representation holds true regardless any regularity assumption
on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$
and show they can be solved without an orthogonal component. To this end, we
extend first existence results for such BSDEs under a general filtration and
then obtain regularity properties such as differentiability for the solution
process.
