Abdelhak GhoulMokhtar HafayedImad Eddine LakhdariShahlar Meherrem2025-10-0620232194-67012194-671X10.1007/s40304-021-00272-5http://dx.doi.org/10.1007/s40304-021-00272-5https://gcris.yasar.edu.tr/handle/123456789/6865In this paper we establish a second-order necessary conditions for stochastic optimal control for jump diffusions. The controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second-order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes.EnglishOptimal control, Stochastic systems with jumps, Pointwise second-order necessary condition, Maximum principle, Variational equationMAXIMUM PRINCIPLE, DIFFERENTIAL-EQUATIONS, SYSTEMS, DELAYArticle