${\mathbb{L}}^p$ $(p>1)$-solutions for BSDEs with jumps and stochastic monotone generator

We study multidimensional discontinuous backward stochastic differential equations in a filtration that supports both a Brownian motion and an independent integer-valued random measure. Under suitable ${\mathbb{L}}^p$-integrability conditions on the data, we establish the existence and uniqueness of ${\mathbb{L}}^p$-solutions for both cases: $p\ge 2$ and $p\in (1,2)$. The generator is assumed to be stochastically monotone in the state variable $y$, stochastically Lipschitz in the control variables $(z,u)$, and to satisfy a stochastic linear growth condition, along with an appropriate ${\mathbb{L}}^p$-integrability requirement.