I have this SDE
$dX_t=[a+bX_t+sX_t(1-X_t)]dt+\frac{1}{2}X_t(1-X_t)dW_t, \, X_0=0,$
where $a,b \in(0,1)$ and let's say that $s$ is a real constant (it's actually a function of $X$, but I think I can get around that and it makes this a bit simpler). I would like to find an explicit solution to this if there is one. So far I have not been very successful in my search. If I eliminate the stochastic part, I get something that is reducible to a linear equation and has a relatively nice explicit solution. But that only gives me a lower bound for the mean of this process and that is much less than I need. Obviously, if I eliminate the quadratic part, this is easily solvable, but again that is not what I need. I welcome any suggestions or references to the literature where a solution could be found. Thanks!
EDIT: Without the stochastic part, I have a Ricatti (http://en.wikipedia.org/wiki/Riccati_equation) equation which can be quite easily solved as I already mentioned. Now, it seems that what I have here is more or less a stochastic version of the Ricatti equation. Does anyone know any good literature on Ricatti SDEs?
EDIT 2: Something one can definitely do is to use an Ito-Taylor expansion. This greatly simplifies things and since the approximation should (I still have to really carefully read through the theory, so don't take my word for granted) weakly converge to the original equation, I can make statements about the distribution of the process $X$ (which is in fact what I actually needed in the first place). This is not an answer to the original question (hence the edit instead of an answer), but perhaps it could be a useful hint to someone who faces a similar problem (or a way for me to let you correct me if I am wrong).