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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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