Backward doubly stochastic differential equations with or without reflection under weak conditions

In this paper, we study the solvability of backward doubly stochastic differential equations (BDSDEs, for short), both with and without reflection, under weak conditions on the generator.

First, when the generator $f$ is of general growth in $y$ and linear growth in $z$, we establish the existence, uniqueness, comparison principle, and the existence of maximal solutions.

Second, when $f$ is of linear growth in $y$ and quadratic growth in $z$ with bounded terminal value, we prove the existence, uniqueness, and comparison principle.

Finally, when $f$ is of general growth in $y$ and quadratic growth in $z$ with bounded terminal value, we prove the existence of maximal solutions.