Discrete stochastic process to stochastic differential equation

Question

I was studying about stochastic processes and got stuck at a question. Let's say, I have a stochastic process like the following,

$X_i(t+1)=\epsilon(X_i + X_j)$

$X_j(t+1)=(1-\epsilon)(X_i + X_j)$

where $\epsilon$ is a uniform random number $\in[0,1]$ and $i,j\in\{1,2,3,...,N\}$. I wanted to know if this process can be formulated in terms of $dX_i/dt$, with an unit of time being $N^2$.

Also is it possible to do this for any discrete stochastic process ? I mean, can any discrete stochastic process be casted into a corresponding stochastic differential equation ?

Thanks in advance

Answer 1

Your model is actually a well-known model in econophysics, it is called the uniform reshuffling model. A discrete time, discrete state space version of this model is studied, for instance, in this paper by Nicolas Lanchier and Stephanie Reed. A continuous time, continuous state space version is studied in a recent paper of myself (joint work with Pierre-Emmanuel Jabin and my Ph.D advisor Sebastien Motsch). In a unpublished work by Roberto Cortez, the author used a coupling approach to study a generic binary exchange model which includes the uniform reshuffling dynamics as a special case, you may found the "SDE representation" (driven by Poisson random measures) in this work. (Spoiler: this is a hard paper to read)