Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 50, pp. 1-19.

Title: Mathematical methods for the randomized non-autonomous Bertalanffy model
       
Authors: Julia Calatayud (Univ. Politecnica de Valencia, Spain)
         Tomas Caraballo (Univ. de Sevilla, Spain)
         Juan Carlos Cortes (Univ. Politecnica de Valencia, Spain)
         Marc Jornet (Univ. Politecnica de Valencia, Spain)

Abstract:
 In this article we analyze the randomized non-autonomous Bertalanffy model
 $$
 x'(t,\omega)=a(t,\omega)x(t,\omega)+b(t,\omega)x(t,\omega)^{2/3},\quad
 x(t_0,\omega)=x_0(\omega),
 $$
 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.