Generalized Fractional Calculus for Gompertz-Type Models

Giacomo Ascione and Enrica Pirozzi
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Giacomo Ascione: Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universitá degli Studi di Napoli Federico II, I-80126 Naples, Italy
Enrica Pirozzi: Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universitá degli Studi di Napoli Federico II, I-80126 Naples, Italy

Mathematics, 2021, vol. 9, issue 17, 1-32

Abstract: This paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by showing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p -moment functions is proved by using generalized Grönwall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve.

Keywords: complete Bernstein function; Gaussian process; linear integral equation; lognormal distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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