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

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 ∈[ t 0 , T ], P ( t 0 , ω )= P 0 ( 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 P 0 to approximate the density function f 1 ( 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 P 0 , and the density of P 0 ,fP 0, 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 ]×Ω), 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. ForfP 0, only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence{ f 1 N ( p , t ) } N = 1 ∞of density functions in terms of expectations that tends to f 1 ( p , t ) pointwise. Numerical examples illustrate our theoretical results.