# Representing a continuous time-inhomogeneous Markov chain by a stochastic integral

I am interested in the following mean-field model introduced in the reference below:

There are $$N$$ particles. At each instant of time, a particle’s state is a particular value taken from the finite state space $$Z = {0, 1, . . . , r − 1}$$. The transition rate for a particle from state $$i$$ to state $$j$$ is governed by mean field dynamics: the transition rate is $$λ_{i,j}(mu_N (t))$$ where $$mu_N (t)$$ is the empirical distribution of the states of particles at time t:
begin{align} mu_N(t)=sum_{i=1}^Ndelta_{x_i} end{align}
The particles interact only through the dependence of their transition rates on the current empirical measure $$mu_N (t)$$ and therefore each particle $$X_n^N(t)$$ is a continuous inhomogeneous-time Markov chain with state-space $$Z$$. My question is:

Is it possible to represent the Markov chains $$X_n^N(t)$$ by a Sde? If so, how can we prove that? I know in the case of homogeneous Markov chain this is possible and we obtain a Sde with respect to Poisson random measure. However, since the inhomogeneity, I don’t know if it is possible to adapt the proof.

Reference: Vivek S. Borkar, Rajesh Sundaresan (2012) Asymptotics of the Invariant Measure in Mean Field Models with Jumps. Stochastic Systems 2(2):322-380. https://doi.org/10.1287/12-SSY064

MathOverflow Asked on November 26, 2021

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