Electron. J. Differential Equations, Vol. 2020 (2020), No. 50, pp. 1-19.

### Mathematical methods for the randomized non-autonomous Bertalanffy model Julia Calatayud, Tomas Caraballo, Juan Carlos Cortes, Marc Jornet

Abstract:
In this article we analyze the randomized non-autonomous Bertalanffy model

where and are stochastic processes and is a random variable, all of them defined in an underlying complete probability space. Under certain assumptions on a, b and , we obtain a solution stochastic process, , both in the sample path and in the mean square senses. By using the random variable transformation technique and Karhunen-Loeve expansions, we construct a sequence of probability density functions that under certain conditions converge pointwise or uniformly to the density function of , . This permits approximating the expectation and the variance of . At the end, numerical experiments are carried out to put in practice our theoretical findings.

Submitted July 20, 2019. Published May 26, 2020.
Math Subject Classifications: 34F05, 60H35, 60H10, 65C30.
Key Words: Random non-autonomous Bertalanffy model; random differential equation; random variable transformation technique; Karhunen-Loeve expansion; probability density function.
DOI: 10.58997/ejde.2020.50

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 Julia Calatayud Instituto Universitario de Matemática Multidisciplinar Universitat Politècnica de València Camino de Vera s/n, 46022 Valencia, Spain email: jucagre@doctor.upv.es Tomás Caraballo Dpto. Ecuaciones Diferenciales y Análisis Numérico Universidad de Sevilla c/ Tarfia s/n, 41012 Sevilla, Spain email: caraball@us.es Juan Carlos Cortés Instituto Universitario de Matemática Multidisciplinar Universitat Politècnica de València Camino de Vera s/n, 46022 Valencia, Spain email: jccortes@imm.upv.es Marc Jornet Instituto Universitario de Matemática Multidisciplinar Universitat Politècnica de València Camino de Vera s/n, 46022 Valencia, Spain email: marjorsa@doctor.upv.es