Here is the list of problems one has to tackle in order to simulate the FP equation.
1) Stability: the finite difference method has to be stable. This can be easily done by using an implicit method.
2) Conservation: the total integral of probability has to be equal to 1. This constraint can be imposed by using probability currents on the right hand side of the FP equation.
3) Positivity: the value of mid point probability can be any combination of two end points of a spatial step. By cleverly using this ambiguity we can insure that probability will never become negative.
Ref to Chang and Cooper for more details.
