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We prove the existence of maximal (and minimal) solution for one-dimensionalgeneralized doubly reflected backward stochastic differential equation (RBSDEfor short) with irregular barriers and stochastic quadratic growth, for whichthe 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 twopredictable barriers $l$ and $u$ on $]0; T]$. This is done without assuming any$P$-integrability conditions and under weaker assumptions on the input data. Inparticular, we construct a maximal solution for such a RBSDE when the terminalcondition $\xi$ is only ${\cal F}_T-$measurable and the driver $f$ iscontinuous with general growth with respect to the variable $y$ and stochasticquadratic growth with respect to the variable $z$. Our result is based on a(generalized) penalization method. This method allow us find an equivalent formto our original RBSDE where its solution has to remain between two new rcllreflecting barriers $\overline{Y}$ and $\underline{Y}$ which are, roughlyspeaking, the limit of the penalizing equations driven by the dominatingconditions assumed on the coefficients. A standard and equivalent form to ourinitial RBSDE as well as a characterization of the solution $Y$ as ageneralized Snell envelope of some given predictable process $l$ are alsogiven.

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