Quadratic BSDEs driven by $Z^2/Y$.
Khaled Bahlali,
Université de Toulon IMATH , France
Abstract.
We introduce a domination argument which roughly expresses that : if we can dominate the parameters (terminal value and driving coefficient) of a Quadratic BSDE from above and from below by those of two BSDEs having an ordered solution, then also our original Quadratic BSDE has a solution. This result will be presented in a general setting, that is without integrability of the solutions. No integrability condition on none of the terminal data of the three involved BSDEs is needed. Neither continuity nor constraints on the growth are required to the dominating coefficients.
Next, we consider a large class of quadratic BSDE which englobe the classical ones and beyond. The domination argument allows us to show that the solvability of this class of BSDEs can be reduced to the solvability of simple BSDEs. This allows to deduce the condition we should impose to the terminal value. The method we propose neither uses a priori estimates nor approximations. A particular developpment concerns the BSDEs which generator $H(t,y,z)$ satisfies $|H(t,y,z)|\leq \alpha_t + \beta_t|y| + \gamma_t|z| + (1/y)|z|^2$.