Doubly reflected BSDEs with stochastic quadratic growth: Around the predictable obstacles

Abstract

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 ξ is only FT−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 allows us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers Y¯ and Y̲ 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.