The case where $b$ or $a$ depends on $Y$ and $Z$

In that case, it is no more possible to approximate (or simulate) the forward process $X$ since its dynamic depends on the solution. In such situations, two different solutions have been proposed:

a - Construct an a priori grid for $X$, possibly based on some a priori on the dynamics of $Y$ and $Z$.

b - Given an a priori solution $Y_{0}$ and $Z_{0}$, approximate (or simulate) the corresponding forward process $X_{0}$ and use the above methodologies to construct the corresponding solution $(Y_{1},Z_{1})$ of the BSDE. Then, use this solution $(Y_{1},Z_{1})$ to approximate (or simulate) the corresponding forward process $X_{1}$ and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement.

Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1].

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