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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?

Savannah
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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?

Ali
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  • If we do not know the exact solution, at least can we say the existence of solution? What I am looking for is the existing theory applicable? – Savannah Jun 19 '22 at 11:58
  • A requirement is for $Y$ to be adapted, if $\xi$ is not "compensating" for the $cW_T$ term then it is not ok – TheBridge Jun 19 '22 at 12:11
  • Okay, I understood the point. The term $\xi$ will have to be random as well, it cannot be an arbitrary constant. – Savannah Jun 19 '22 at 14:28
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    Noone is suggesting that $\xi$ should be constant. That would not help you anyway unless $c=0$. You should infer from what I wrote that a solution can exist only when $\xi$ has a form like $a+cW_{T}$ (still very much a random variable). – Ali Jun 19 '22 at 16:57
  • Thank you Ali I understood. – Savannah Jun 20 '22 at 12:51