Solution of a particular Backward SDE I have been reading about backward SDEs and their existence-uniqueness result. The results are available for equations of the form $$ -dY_t = f(t,\omega, Y_t,Z_t)\,dt- Z_t\,dW_t, \quad Y_T= \xi.$$
A solution is a pair $(Y_t,Z_t)$ such that $Z_t$ is predictable and we must have the relation $Y_t= \xi + \int^T_t f\,ds-\int^T_tZ_s\,dW_s$. So my question is if we have a SDE of the form $-dY_t = f(Y_t)\,dt- cdW_t,\,\, Y_T= \xi$, i.e. the diffusion coefficient is constant, can we still talk about existence and uniqueness? Will $(Y_t, c)$ be a pair of solution or will there be some issue?

Suppose for simplicity that $f=0$. With $Z_{s}=c$ we get
\begin{equation*}
Y_{t}=\xi-\int_{t}^{T} Z_{s}dW_{s}=\xi-c\left(W_{T}-W_{t}\right).
\end{equation*}
Unless $\xi$ looks something like $a +cW_{T}$ so that terms cancel, we end up with a $Y_{t}$ depending on the increment $W_{T}-W_{t}$. You haven't actually written anything (like the word "adapted") to exclude that possibility, but probably you wanted to?
