Multivalued backward stochastic differential equations with jumps and moving boundary

Abstract

We prove existence and uniqueness for a one-dimensional multivalued backward stochastic differential equation with jumps. The equation involves a time-indexed family of maximal monotone operators $k_t(\cdot )$ associated with increasing functions $k(t,\cdot )$ taking values in ${\mathbb{R}}_{-}$ and having domains that are intervals with time-dependent boundaries. Existence is obtained by a penalization method under a Lipschitz condition on the driver in $(y,z)$, a monotonicity condition in the jump parameter $\ddot{I}ˆ$, square-integrability of the terminal condition and the driver, and local-in-time integrability conditions on $k(\cdot ,y)$. We also address the extension to the case where the operators $k_t(\cdot )$ act on unbounded intervals.