Abstract The purpose of this study is to develop a stochastic Newmark integration principle based on an implicit stochastic Taylor (Ito–Taylor or Stratonovich–Taylor) expansion of the vector field. As in the deterministic case, implicitness in stochastic Taylor expansions for the displacement and velocity vectors is achieved by introducing a couple of non-unique integration parameters, α and β. A rigorous error analysis is performed to put bounds on the local and global errors in computing displacements and velocities. The stochastic Newmark method is elegantly adaptable for obtaining strong sample-path solutions of linear and non-linear multi-degree-of freedom (m.d.o.f.) stochastic engineering systems with continuous and Lipschitz-bounded vector fields under (filtered) white-noise inputs. The method has presently been numerically illustrated, to a limited extent, for sample-path integration of a hardening Duffing oscillator under additive and multiplicative white-noise excitations. The results are indicative of consistency, convergence and stochastic numerical stability of the stochastic Newmark method (SNM).