Continuous And Discrete Stochastic Models Related Through Finite Difference Forward Kolmogorov Equations

Discrete and continuous stochastic models that have approximate probability densities are identified through finite difference approximations of forward Kolmogorov equations (FKEs). Finite difference constructions of the FKE of an Itˆo stochastic differential equation (SDE) are shown to be identical to the FKEs of certain stochastic models that are discrete in time and state. This identification yields a family of discrete stochastic models with probability densities that approximate the probability density of the SDE model. Additionally, the FKE of a discrete stochastic model is shown to be equal to a finite difference equation of the FKE of a continuous stochastic model. This work gives greater understanding of how discrete and continuous stochastic models are related. Several examples in natural sciences illustrate joint application of realistic discrete and continuous stochastic models.