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.



Finding a stochastic differential equation as limit of a discrete stochastic process
Here is a sketch of how you might approach this. Let
$$
 \xi\_j = \frac{\kappa(I\_{j-1} - X\_{j-1})}
 {\sqrt{X\_{j-1}(1 - X\_{j-1})}}.
 $$
The two "hard" results that must be proven are: (1) For each $t$,
$$
 \lim\_{\kappa\to0} E\big[\max\{|\xi\_j|: 1 \le j
 \le \lfloor\kappa^{-2}t\rfloor\}\big] = 0,
 $$
and (2) for each $t$,
$$
 \sum\_{j=1}^{\lfloor\kappa^{-2}t\rfloor} \xi\_j^2 \to t,
 $$
in probability as $\kappa\to0$. The rest of the proof would then be the following "soft" argument based on general theory.


First, let $W^\kappa(t)=\sum\_{j=1}^{\lfloor\kappa^{-2}t\rfloor} \xi\_j$. Using the two results above, one can use the martingale central limit theorem (Theorem 7.1.4 in [Ethier & Kurtz](https://rads.stackoverflow.com/amzn/click/com/047176986X)) to prove that $W^\kappa\Rightarrow W$, where $W$ is a standard Brownian motion.


Next, we take the difference equation which defines the sequence $\{X\_n\}$ and rewrite it as an integral equation. More specifically, if we define $X^\kappa(t)=X\_{\lfloor\kappa^{-2}t\rfloor}$, then we may write
$$
 X^\kappa(t) = X\_0 + \int\_0^t \sqrt{X^\kappa(s-)(1 - X^\kappa(s-))}\\,dW^\kappa(s).
 $$
There is nothing deep here, just a change of notation, really.


Finally, we use Theorem 5.4 in [Kurtz & Protter](https://projecteuclid.org/journals/annals-of-probability/volume-19/issue-3/Weak-Limit-Theorems-for-Stochastic-Integrals-and-Stochastic-Differential-Equations/10.1214/aop/1176990334.full) to prove that $(X\_0,X^\kappa,W^\kappa) \Rightarrow(X\_0,X,W)$, where $X$ is the unique strong solution to $dX=\sqrt{X(1-X)}\\,dW$, $X(0)=X\_0$.


A watered-down version of Theorem 5.4 in Kurtz & Protter is available as Theorem 2.3 in [these lecture notes](https://math.swansonsite.com/instructional/skor_lemmas.pdf). This version is sufficient for your purposes, and it may be easier to digest. Also, to use this theorem, you must show that, for every version of $(X\_0,W)$, the limiting SDE has a unique strong solution for all time. This follows, for example, from Proposition 5.2.13, Theorem 5.5.4, and Corollary 5.3.23 in [Karatzas & Shreve](https://rads.stackoverflow.com/amzn/click/com/0387976558).


