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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 Markovmartingale with respect to a Markov filtration $\mathcal{F}$ admits an exactintegral representation with respect to $M$, that is, without any orthogonalcomponent. This representation holds true regardless any regularity assumptionon $F$. 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, weextend first existence results for such BSDEs under a general filtration andthen obtain regularity properties such as differentiability for the solutionprocess.

Weak martingale representation for continuous Markov processes and application to quadratic growth BSDEs