On the orthogonal component of BSDEs in a Markovian setting

In this Note we consider a quadratic backward stochastic differential equation (BSDE) driven by a continuous martingale $M$ and whose generator is a deterministic function. We prove (in Theorem \ref{theorem:main}) that if $M$ is a strong homogeneous Markov process and if the BSDE has the form \eqref{BSDE} then the unique solution $(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the orthogonal martingale $N$ is equal to zero showing that in a Markovian setting the "usual" solution $(Y,Z)$ has not to be completed by a strongly orthogonal even if $M$ does not enjoy the martingale representation property.