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 ,

becomes with probability

and becomes with probability

we have and with denoting the variance of with known.

Numerically i found that but this is an aproximation because there should not be any chance of 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 and not