Multivalued backward stochastic differential equations with jumps and moving boundary
Badr Elmansouri, Anas Ouknine, Youssef Ouknine
TL;DR
This work analyzes one-dimensional multivalued backward SDEs with Poisson jumps and a moving boundary, where the state $Y_t$ remains in the moving domain $[a_t,\infty)$ and is driven by a maximal monotone operator $k_t(\cdot)$ tied to an increasing function $k(t,\cdot)$. Existence and uniqueness are established via a penalization (Yosida) approach under Lipschitz in $(y,z)$ and monotonicity in $\psi$, together with square-integrability of data and local-in-time integrability on $k(\cdot, y)$; a comparison principle is also proven. The paper first handles the case where the operator graphs lie in $\mathbb{R} \times \mathbb{R}_-$, then extends to unbounded domains using localization and concatenation. An extension to the general case with graphs in $\mathbb{R} \times \mathbb{R}$ is provided via truncation and a concatenation argument across stopping times, yielding a global solution under a structural assumption (C). Overall, the results furnish a rigorous framework for MBSDEs with jumps and moving barriers, with potential applications in stochastic control and finance under state constraints and time-evolving domains.
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 $ψ$, 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.