Generalized Fractional Calculus for Gompertz-Type Models

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.