Improving the approximation of the probability density function of random nonautonomous logistic-type differential equations

Abstract

[EN] In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P-'(t,omega)=A(t,omega)(1-P(t,omega))P(t,omega), t is an element of[t(0),T], P(t(0),omega)=P-0(omega), where omega is any outcome in the sample space omega. In the recent contribution [Cortes, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121-138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P-0 to approximate the density function f(1)(p,t) of P(t): A(t) is expressed as a Karhunen-Loeve expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P-0, and the density of P-0, fP0, is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L-2([t(0),T]x omega), so that we include other expansions such as random power series. We only require absolute continuity for P-0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For fP0, only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence {f1N(p,t)}N=1 infinity of density functions in terms of expectations that tends to f(1)(p,t) pointwise. Numerical examples illustrate our theoretical results.