| Abstract: |
| This talk establishes the existence of maximal (and minimal) solution for one-dimensional generalized reflected backward stochastic differential equation (RBSDE for short). The equation features irregular barriers and a driver with stochastic quadratic growth. The solution $Y$ is constrained to remain above rcll barriers $L$ and $U$ on $[0, T[$, while its left limit $Y_-$ must stay above predictable barriers $l$ and $u$ on $]0, T]$. This result is achieved without assuming any $\mathbb{P}$-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 $\mathcal{F}_T$-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 proof is based on a generalized penalization method. Furthermore, we present a standard and equivalent formulation of the original RBSDE and characterize the solution $Y$ as the generalized Snell envelope of a specific predictable process $l$. |
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