The Stability and Convergence of Carathéodory Approximate Solutions for Nonlinear SDEs With G‐Lévy Jumps

ABSTRACT The objective of this article is to provide an analysis of solutions for stochastic differential equations (SDEs) that incorporate G‐Lévy jumps using the Carathéodory approximation technique. The existence theory for SDEs driven by G‐Lévy jumps has been established, even in cases where non‐Lipschitz conditions are present. It has been investigated that the Carathéodory approximate solutions are bounded and converge to a solution of SDEs with G‐Lévy jumps. We have derived that SDEs featuring G‐Lévy jumps exhibit at most a single solution. Various well‐established tools, including the Burkholder–Davis–Gundy (BDG) inequality, the Borel–Cantelli lemma, and the Bihari–LaSalle inequality are used to provide estimations for these findings. Furthermore, the mean square stability of these equations has been successfully evaluated.