Hi, i hope that i am at the correct category.

This question is bugging me for too long, so i hope someone will be able to help.

I have a discrete stochastic process defined as

Let $\displaystyle 0\le k, a_0\le 1$,

$\displaystyle a_{n+1}$ becomes $\displaystyle a_n+k(1-a_n)$ with probability $\displaystyle a_n$

and $\displaystyle a_{n+1}$ becomes $\displaystyle a_n-k a_n$ with probability $\displaystyle 1-a_n$

we have $\displaystyle E(a_{n+1}|a_n)=a_n$ and $\displaystyle V(a_{n+1}|a_n)=k^2 a_n(1-a_n)$ with $\displaystyle V(a_{n+1}|a_n)$ denoting the variance of $\displaystyle a_{n+1}$ with $\displaystyle a_n$ known.

Numerically i found that $\displaystyle a_t \backsimeq N[a_0,k^2 a_0(1-a_0)t]$ but this is an aproximation because there should not be any chance of $\displaystyle a_t$ being outside of [0,1].

So what i am trying to do is to construct the continuous process which is the limit of the above discrete one. Much like random walk and weiner process.

I would also be content if i was able to construct the corresponding SDE of the process, or the PDE for the evolution of the probability density. In short, any help is more than welcome.

EDIT:The probability of increase is $\displaystyle a_n$ and not $\displaystyle a$