Improving the approximation of the probability density function of random nonautonomous logistic-type differential equations
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https://doi.org/10.1002/mma.5834 |
Metadatos
Título
Improving the approximation of the probability density function of random nonautonomous logistic-type differential equationsFecha de publicación
2019-08-16Editor
John Wiley & Sons, Ltd.ISSN
0170-4214; 1099-1476Cita bibliográfica
Calatayud, J., Cortés, J. C., & Jornet, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations. Mathematical Methods in the Applied Sciences, 42(18), 7259-7267.Tipo de documento
info:eu-repo/semantics/articleVersión de la editorial
https://onlinelibrary.wiley.com/doi/10.1002/mma.5834Versión
info:eu-repo/semantics/publishedVersionPalabras clave / Materias
Resumen
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, ... [+]
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 [-]
Publicado en
Mathematical Methods in the Applied Sciences, Vol. 42, Issue18 (December 2019)Entidad financiadora
Secretaría de Estado de Investigación, Desarrollo e Innovación | Universitat Politècnica de València
Código del proyecto o subvención
MTM2017-89664-P
Título del proyecto o subvención
Programa de Ayudas de Investigación y Desarrollo
Derechos de acceso
© 2019 John Wiley & Sons, Ltd.
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