Kumaraswamy diffusion processes

Abstract

This paper introduces a set of bounded continuous-time diffusion models with standard and modified Kumaraswamy distributions as their invariant and limiting distributions. Both Kumaraswamy distributions, including their reflected versions, are considered. The models are constructed through transformations of Jacobi diffusions, with two distinct construction methods giving rise to type I and type II diffusions. The resulting models have distinct pathwise properties while sharing the same invariant distribution and provide flexible and tractable alternatives for stochastic modeling of bounded processes in a range of applications. Explicit eigenfunction expansions for the transition densities and conditional moments are derived for all four model classes. Numerical examples illustrate the properties of the proposed diffusions, and an application to statistical risk monitoring in the context of water reservoir management is presented.