# Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers

## TL;DR

This paper investigates a class of anticipated backward stochastic differential equations with jumps, allowing complex growth conditions on the driver, and proves existence and uniqueness of solutions in both Markovian and non-Markovian contexts.

## Contribution

It introduces a framework for ABSDEs with jumps and complex growth drivers, establishing existence, uniqueness, and regularity results under new structural assumptions.

## Key findings

- Existence and uniqueness of solutions for ABSDEs with jumps and quadratic-exponential drivers.
- Regularity properties of the solution components in the Markovian case.
- Extension of ABSDE theory to drivers with linear, quadratic, and exponential growth.

## Abstract

In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple $(Y,Z,\psi)$ where $Y$ is a semimartingale, and $(Z,\psi)$ are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of $Y$'s future paths, as well as quadratic and exponential growth on the spot values of $(Z,\psi)$, respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on $(Z,\psi)$ with respect to the forward process are also obtained.