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The rigorous linking of exact stochastic models to mean-field approximationsis studied. Starting from the differential equation point of view thestochastic model is identified by its Kolmogorov equations, which is a systemof linear ODEs that depends on the state space size ($N$) and can be written as$\dot u_N=A_N u_N$. Our results rely on the convergence of the transitionmatrices $A_N$ to an operator $A$. This convergence also implies that thesolutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODEcan be easily used to derive simpler mean-field-type models such that themoments of the stochastic process will converge uniformly to the solution ofappropriately chosen mean-field equations. A bi-product of this method is theproof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turnsout that the proof holds for cases that are slightly more general than theusual density dependent one. Moreover, for Markov chains where the transitionrates satisfy some sign conditions, a new approach for proving convergence tothe mean-field limit is proposed. The starting point in this case is thederivation of a countable system of ordinary differential equations for all themoments. This is followed by the proof of a perturbation theorem for thisinfinite system, which in turn leads to an estimate for the difference betweenthe moments and the corresponding quantities derived from the solution of themean-field ODE.

Differential equation approximations of stochastic network processes: an operator semigroup approach