On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale MM. We prove (in Theorem 3.2) that if MM is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution (Y,Z,N)(Y,Z,N) of the BSDE is reduced to (Y,Z)(Y,Z), i.e.   the orthogonal martingale NN is equal to zero, showing that in a Markovian setting the “usual” solution (Y,Z)(Y,Z) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if MM does not enjoy the martingale representation property.