In the last decades there has been a growing interest towards the problem of the approximation of stochastic systems, shared by many different disciplines.
The Finite State Expansion, recently proposed, is an hybrid approximation method for the dynamics of Chemical Reaction Networks that aims at finding a compromise between a microscopic probabilistic description of a part of the system and a macroscopic deterministic approximation of the rest of it.
In this talk we will deal the problem of its convergence, which reduces to showing that a sequence of finite systems of Ordinary Differential Equations converges to an infinite-order given system.
From the analytical point of view this problem presents some challenging aspects that will be analyzed, together with possible approaches to their resolution.
In particular we will present a first possible proof, based on a conjecture, that gives some interesting insights into the mathematical relations behind the algorithm.
Moreover, we will show how the same problem can be rephrased in terms of stability for perturbed systems of ODEs, highlighting the critical points that make the proof intrinsically difficult.