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

This paper investigates the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕃</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {Lᵖ} -solutions ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {p∈(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕃</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {Lᵖ} framework. Building upon this fundamental result, we prove the existence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕃</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {Lᵖ} -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.