We have presented a new numerical method to find the densities of stochastic differential equations, and several related stochastic integrals. We have coined the term “stochastic calculus of standard deviations” for this new technology. We invite interested readers to read our research paper on SSRN titled,”Stochastic Calculus of Standard Deviations: An Introduction.” We have presented this method as a faster alternative to mote carlo simulations. We are confident that our method and its derivatives will have wide spread applications in finance and management, mathematics, statistics, physics, chemistry and biological sciences among other disciplines.
We have presented a new method to calculate the conditional distribution of moments of integrals of functions of a random process governed by a given stochastic differential equation.The method can easily be extednded to find the conditional distributions of various path dependent functions of the stochastic process. The paper can be downloaded here.
We have presented a new numerical method to generate transition probabilities of mean reverting SDEs using Girsanov Theorem. This makes generation of transition probabilities possible even on grids where numerical solution of PDE would most likely break down.The technique can easily be extended to other stochastic differential equations with more complex dynamics that are not necessarily mean reverting. We hope that this new method will be a good step towards the future wide spread use of measure theory in mathematical finance and other related statistical disciplines.The paper on this method will soon appear on SSRN. The experimental code for this method can be
We have extended our earlier work with stochastic calculus of standard deviations and now we can calculate densities of stochastic differential equations with a very general drift and volatility structure. We also calculate densities of time integrals of various functions of stochastic processes governed by a given stochastic differential equation. The distributions of these time integrals of stochastic processes remain extremely accurate for short periods of time like a quarter year and can be used with Girsanov method of simulation with transition probabilities for highly accurate long horizon simulation. These stochastic time integrals can also be used to calculate densities of more complex functions of stochastic processes for example we can find highly accurate distributions of various moments of functions of a stochastic processes by employing these stochastic integrals in conjunction with transition probabilities. The experimental code for this method can be downloaded here. We will follow with more code for this method in about two weeks that will map the densities of stochastic integrals from standard deviations grid to a fixed grid so that it could be directly used in the transition probabilities framework.