I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm.

Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and

$X_{n+1}=X_n+\kappa(I_n-X_n)$

with $I_n=1$ with probability $X_n$ and $I_n=0$ with probability $1-X_n$.

In other words the $X_n$ decreases with probability $1-X_n$ by $\kappa X_n$ and increases with probability $X_n$ by $\kappa(I_n-X_n)$ so $E[X_{n+1}]=X_n$.

The point is that $\kappa$ can be arbitrarily small so we can take its limit to $0$ while decreasing linearly the time step. This naturally should give an SDE (in this case I would expect it to be non-linear). So my question is how can one find this SDE or the PDE that gives the probability density.

I should add that for short times it looks like a random walk (which is expected I guess) with the variance being proportional to $\kappa^2 t^2 X_0(1-X_0)$, with $t$ small. However since $X_n\in[0,1]$, $1$ is an upper bound for the variance.