How can I obtain from a differential equation a stochastic version? Suppose $\frac{dx}{dt}=ax+b$ and then assume that $a=c+g$ where $g$ is a Wiener process. 

I think that 'Stochastic Differential Equations' by Bernt Oksendal is a good book about SDE.
According your equation, lets rewrite it:
$$dx = ax\,dt +b\,dt$$
and add noise
$$dx = (cx+b)\,dt + xg\,dt$$
If $g$ is a Wiener process, then $g\,dt$ is a Brownian motion, so your SDE equation is
$$dX_t(\omega) = (cX_t(\omega)+b)\,dt + X_t(\omega)dB_t(\omega)$$
