Solutions and simulations of some one-dimensional stochastic differential equations

We consider a one dimensional SDE dX t = I?(X t )dt + I?(X t )dB t . We give a new general formula for solutions that involves solving an associated ordinary differential equation. Explicit solutions are obtained in cases where the ODE has such. This recovers linear case but also some non-linear cases. In any case our approach leads to a new simulation scheme that returns positive values for processes on \hskip 0.02in \hbox \rm I \hskip -.02in \hbox \rm R ^+ , which is advantageous when modelling prices or rates.