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It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.)

How can I solve $dY_t=Z_tdW_t+ a(Y_t)^+dt +bZ_tdt $ with $Y_T=\xi$ ?

($(Y_t)^+$ means the positive part of $Y_t$)

This question is very important for me. I have got stucked in it for more than one week, after trying several methods. I will be very grateful if anyone can give me some hint about possible ways to solve it! Please, any suggestion is very important for me.

I am also interested in solving $dY_t=Z_tdW_t+ a(Y_t)^+dt +b(Z_t)^+dt $ with $Y_T=\xi$ ?

I am a starter in Backward SDE and wish to explore deeper in how to solve different BSDEs. Can anyone suggest me some related books?

Thank you so much!!

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The easiest way is to assume Y is positive and solve it. If the solution is positive, then it is the actual solution. Another way is that if the BSDE is Markovian, you can solve the corresponding PDE(Feynman-Kac formula). In general, the solution should depend heavily in choosing $\xi$. Therefore, one may not get general solution for any $\xi$.

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