Coupled stochastic systems of Skorokhod type: well-posedness of a mathematical model and its applications

The paper claims existence, strong uniqueness, and parameter dependence for a reflected SDE with a finite-activity Poisson jump term under assumptions (A1)–(A6) (Theorems 3.1–3.3) . However, its solution concept explicitly requires X and the regulator Φ to be continuous processes (Definition 2.2) , while the equation includes a pure-jump term ∫ρ(X(s),y)ν(dy,ds) that is exactly a compound Poisson integral (hence cadlag with jumps) . The existence proof incorrectly invokes a Skorokhod map formulated on continuous paths to treat drivers with jumps and, in the key Lemma 3.5, drops the Poisson term in the limiting equation altogether (compare (3.37)–(3.38) with (3.39)–(3.40)) . The uniqueness proof similarly ignores the jump contribution and mis-specifies notation, providing no control of the Poisson part . By contrast, the model solution correctly treats the process as cadlag, uses Skorokhod maps for cadlag inputs, controls the Poisson part via BDG and Itô’s formula for jump semimartingales, obtains pathwise uniqueness by a boundary barrier estimate, and then invokes Yamada–Watanabe for semimartingale-driven equations, yielding strong existence and uniqueness under the stated assumptions.