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Let D be an open convex set in ${ℝ}^d$ and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: $X_t=H_t+{\int }_0^t⟨F{\left(X\right)}_{s-},d{Z}_s⟩+K_t$, t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.