Here is new version of "Introduction to Stochastic Calculus of Standard Deviations." Now the evolution of distributions of SDEs is given in terms of change of variance of noise driving the sde. Here is the address to read or download it.https://docs.google.com/file/d/0B1UoJb9 ... haringHere
is the abstract.Every density produced by an SDE which employs normal random variables for its simulation is a linear or mostly non-linear transformation of normal random variables. We find this transformation in case of a general SDE by taking into account how variance evolves in that certain SDE. We find Jacobian of this transformation with respect to normal density and employ change of variables formula for densities to get the density of simulated SDE. Briefly in our method, domain of the SDE is divided into standard deviation fractions that expand or contract as the variance increases or decreases such that probability mass within each SD fraction remains constant. Usually 200-400 SD fractions are enough for desired accuracy. Within each SD fraction stochastic integrals are evolved independently of other SD fractions. The work for each step is roughly the same as that of one step in monte carlo but since SD fractions are only a few hundred, this technique is much faster than monte carlo. Since this technique is very fast, we are confident that it will be the method of choice to evolve distributions of SDEs as compared to monte carlo simulations and partial differential equations.