In my last question, Jose and I discussed about the solution to a linear backward SDE. I followed his steps and made it clear.
Besides, I read a paper from Professor Peng talk about the linear Backward SDE. In that paper, Professor Peng gave the result of the linear backward SDE:
for $g(t,Y_t,Z_t) = a_tY_t + b_tZ_t + c_t$ is a linear function, the solution of $Y_t$ can be write as the following form:
$$Y_t = E[X_T\xi + \int _t^Tc_sX_sds|\mathscr{F}_t]$$
Where $X_t$ is the solution of the following SDE:
$$dX_s=a_sX_sds + b_sX_sdB_s$$
Where $s\in[t,T]$ and $X_t = 1$
Here comes the problem: how can we get the form of the solution of $Y_t$?
What I know is: $$X_tY_t = E[X_T\xi-\int_t^T X_sc_sds|\mathscr{F}_t]$$ But how to transform it into the solution of $Y_t$?