A Fractional Stochastic Gompertz-Type Model Induced By Bernstein Functions

FD2W01 - Deterministic and stochastic fractional diï¬€erential equations and jump processes

In [1] we studied a class of linear fractional-integral stochastic equations, for which an existence and uniqueness result of a Gaussian solution was proved. We used such kind of equations to construct fractional stochastic Gompertz models, in such a way we included the fractional Gompertz curves previously introduced in [3] and [4]. Then, in [2], we focus on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Specifically, we introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of the solutions. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. A fractional rate process and a generalization of lognormal distrubution are also provided. (This is a joint work with Giacomo Ascione.) References1 Ascione, G.; Pirozzi, E. On the Construction of Some Fractional Stochastic Gompertz Models. Mathematics 8, 60 (2020) https://doi.org/10.3390/math80100602 Ascione, G.; Pirozzi, E. Generalized Fractional Calculus for Gompertz-Type Models. Mathematics 2021, 9, 2140. https://doi.org/10.3390/math91721403 Bolton, L.; Cloot, A.H.; Schoombie, S.W.; Slabbert, J.P. A proposed fractional-order Gompertz model and its application to tumour growth data. Mathematical medicine and biology: a journal of the IMA 32 , (2014), 187–209.[4] Frunzo, L.; Garra, R.; Giusti, A.; Luongo, V. Modeling biological systems with an improved fractional Gompertz law. Communications in Nonlinear Science and Numerical Simulation, 74, (2019), 260–267.

This talk is part of the Isaac Newton Institute Seminar Series series.