I am interested by this problem
Find a solution to this backward stochastic differential equation :
$\ y(t) = (ry(t) + az(t))*dt + z(t)dW_t$
with the terminal condition
$y(T) = \xi$
with $\xi$ is a square integrable, $F_T$ measurable random variable in a filtered probability space and $W$ is a one dimensional brownian motion. in addition $r$ and $a$ are constant.
What I have tried for the moment is introduction $x(t) = y(t)e^{-rt}$ to apply Ito's lemma, in order to get rid of the term in ry(t) in the left-hand side of the equation. In order to be able to apply the Martingale representation, but I still have a term depending on dt.
If you have some way to solve this kind of equation I would be grateful. (Note that we can solve explicitly $y(t)$ but we can't I think have an explicit solution for z(t) only an implicit form thanks to the martingale representation.
In advance thank you very much.