Closed-form formulas for conditional moments of inhomogeneous Pearson diffusion processes

摘要

Diffusion models have been thoroughly studied for their use in seeking stochastic differential equation (SDE) solutions and investigating their properties, such as moments and conditional moments, which play significant roles in many real-world applications and are especially beneficial for estimating parameters. In fact, these moments can be directly calculated by applying the transition probability density function (PDF), which is often unknown or unavailable in closed form; the formulas for the conditional moments of the SDE may be unavailable in closed form, as well. In this work, we studied an extended case of Pearson diffusion processes, which are a class of diffusions that have squared diffusion coefficients with time-dependent parameter functions. A complete investigation was carried out for both light-and heavy-tailed Pearson diffusion processes, including Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, Fisher-Snedecor, reciprocal gamma, and Student. We introduce a simple but novel approach to closed-form formulas for conditional moments of inhomogeneous Pearson diffusion processes. The approach does not require any knowledge of eigenfunctions or the transition PDF. In each class of stationary distributions reduced from Pearson diffusions, the formula is explored and presented in a concise form. The closed-form formulas obtained are also numerically validated by MC simulations. (C) 2021 Elsevier B.V. All rights reserved.