Fractional Cauchy problems for time-changed Pearson diffusions: Explicit solutions and conditional moments

Phiraphat Sutthimat and Phisitphong Ketrat

Chaos, Solitons & Fractals, 2026, vol. 210, issue P2

Abstract: This paper studies a class of fractional Cauchy problems associated with time-changed Pearson diffusions under a Caputo derivative of order α∈(0,1]. Focusing on time-homogeneous Pearson diffusions, we show that the governing problem can be reduced to a finite coupled system of linear Caputo fractional differential equations with polynomial structure. By combining Laplace transform techniques with confluent partial-fraction expansions, we derive explicit solution representations in terms of Mittag–Leffler functions, with Prabhakar functions naturally arising from the inverse Laplace transforms of higher-order pole terms. These general formulas are then used to obtain closed-form conditional moments for the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, and Jacobi diffusions time-changed by the inverse of a stable subordinator. The obtained results extend the classical conditional moments to the fractional setting and recover the standard formulas when α=1. To validate the analytical expressions, we perform Monte Carlo simulations based on first-passage inversion of the stable subordinator and compare the numerical estimates with the derived formulas over a range of parameter values. The numerical results, assessed through root mean square error values, confidence intervals, and Monte Carlo comparisons with N=3×104 simulated paths, provide numerical validation of the derived analytical formulas over the tested parameter settings. The present study provides an explicit and computationally effective approach to a class of fractional diffusion problems for which closed-form solutions are usually difficult to obtain.

Keywords: Conditional moment; Fractional Cauchy problem; Fractional Pearson diffusion; Fractional derivative; Subordinator (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p2:s0960077926008581

DOI: 10.1016/j.chaos.2026.118717

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