Stochastic Calculus of Standard Deviations: An Introduction

Every density produced by an SDE which employs normal random variables for its simulation is either linear or non-linear transformation of the normal random variables. We find this transformation in case of a general SDE by taking into account how the variance evolves in that certain SDE. We map the domain of the normal distribution into the domain of the SDE by using the algorithm given in the paper which is based on how the variance grows in the SDE. We find the Jacobian of this transformation with respect to normal density and employ a change of variables formula for densities to get the density of simulated SDE. Briefly in our method, domain of the normal distribution is divided into equal subdivisions called standard deviation fractions that expand or contract as the variance increases or decreases such that probability mass within each SD fraction remains constant. Usually 300-500 SD fractions are enough for desired accuracy. Within each normal SD fraction, stochastic integrals are evolved/mapped from normal distribution to distribution of SDE based on change of local variance 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 and are independent of each other, this technique is much faster than the monte carlo simulation. Since this technique is very fast, we are confident that it will be the method of choice to evolve distributions of the SDEs as compared to the monte carlo simulations and the partial differential equations.