Calculation of Transition Probabilities of SDEs Only from the Knowledge of Marginal Probabilities Using the CDF-Equivalent Brownian Motion Method

Abstract

In this article, we show how to calculate the conditional and transition probabilities of any SDE between two different points across time only from the knowledge of their marginal probabilities on these two time grids. Briefly, we construct CDF-Equivalent standard Brownian motion grids from the knowledge of marginal probabilities of the SDE. The solution of conditional and transition probability of the SDE is the found from the solution of transition probabilities between the corresponding nodes of the CDF-Equivalent standard Brownian motion grids which can be easily calculated. What is the most interesting is that we do not need to know the SDE dynamics for calculation of conditional probabilities and the method requires only the knowledge of marginal probabilities. We can, in fact, use this method to determine the SDE that is best suited as a data generating process.