Mathematical methods for the randomized non-autonomous Bertalanffy model

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.

Keywords

• random non-autonomous bertalanffy model
• random differential equation
• random variable transformation technique
• karhunen-loeve expansion
• probability density function