On the 𝕃 p -solution for BSDELs with continuous or left continuous coefficient

This paper investigates the 𝕃 p {\mathbb{L}^{p}} -solutions ( p ∈ ( 1 , 2 ) {p\in(1,2)} ) for backward stochastic differential equations driven by Teugels martingales associated with a Lévy process (BSDELs). While existence and uniqueness were established by El Jamali (2023) under Lipschitz conditions, the present work focuses on establishing a comparison theorem specifically tailored for the 𝕃 p {\mathbb{L}^{p}} framework. Building upon this fundamental result, we prove the existence of 𝕃 p {\mathbb{L}^{p}} -solutions for BSDELs under significantly weakened assumptions on the coefficient. Specifically, we consider cases where the generator is either continuous with linear growth, or satisfies a monotonicity condition-being increasing (resp. decreasing)-combined with left (resp. right) continuity. These results extend the theoretical reach of BSDEL theory to a broader class of stochastic processes and coefficients.