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 $a(t,\omega)$ and $b(t,\omega)$ are stochastic processes and $x_0(\omega)$ is a random variable, all of them defined in an underlying complete probability space. Under certain assumptions on a, b and $x_0$, we obtain a solution stochastic process, $x(t,\omega)$, 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 $x(t,\omega)$, $f_{x(t)}(x)$. This permits approximating the expectation and the variance of $x(t,\omega)$. 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

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