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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 has to remain between two rcll barriers and on , and its left limit has to stay respectively above and below two
predictable barriers and on . This is done without assuming any
-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 is only measurable and the driver is
continuous with general growth with respect to the variable and stochastic
quadratic growth with respect to the variable . 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 and 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 as a
generalized Snell envelope of some given predictable process are also
given.Comment: 21 page