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A Fractional Stochastic Gompertz-Type Model Induced By Bernstein Functions

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FD2W01 - Deterministic and stochastic fractional differential 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. Mathematics 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.

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