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

Abstract

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P ′ (t, 𝜔�) = A(t, 𝜔�)(1 − P(t, 𝜔�))P(t, 𝜔�), t ∈ [t0, T], P(t0, 𝜔�) = P0(𝜔�), where 𝜔� is any outcome in the sample space Ω. In the recent contribution [Cortés, 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 P0 to approximate the density function f1(p, t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P0, and the density of P0, 𝑓�P0 , 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 L2([t0, T] × Ω), so that we include other expansions such as random power series. We only require absolute continuity for P0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For 𝑓�P0 , only almost everywhere continuity and boundedness on (0, 1) are needed. We construct an approximating sequence {𝑓� N 1 (p, t)}∞ N=1 of density functions in terms of expectations that tends to f1(p, t) pointwise. Numerical examples illustrate our theoretical results