Doubly Reflected BSDEs With Stochastic Quadratic Growth: Around The Predictable Obstacles

We prove the existence of maximal (and minimal) solution for one-dimensional generalized doubly reflected backward stochastic differential equation (RBSDE for short) with irregular barriers and stochastic quadratic growth, for which the solution $Y$ has to remain between two rcll barriers $L$ and $U$ on $[0; T[$, and its left limit $Y_-$ has to stay respectively above and below two predictable barriers $l$ and $u$ on $]0; T]$. This is done without assuming any $P$-integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a RBSDE when the terminal condition $\xi$ is only ${\cal F}_T-$measurable and the driver $f$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. Our result is based on a (generalized) penalization method. This method allow us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers $\overline{Y}$ and $\underline{Y}$ which are, roughly speaking, the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. A standard and equivalent form to our initial RBSDE as well as a characterization of the solution $Y$ as a generalized Snell envelope of some given predictable process $l$ are also given.