@Article{JCM-43-1219,
author = {Schurz , Henri},
title = {Logistic Stochastic Differential Equations with Power-Law},
journal = {Journal of Computational Mathematics},
year = {2025},
volume = {43},
number = {5},
pages = {1219--1237},
abstract = {<p style="text-align: justify;">An analysis of logistic stochastic differential equations (SDEs) with general power-law
and driven by a Wiener process is conducted. We prove existence of unique, strong Markovian, continuous solutions. The solutions live (a.s.) on bounded domains $D = [0, K]$ required by applications to biology, ecology and physics with nonrandom threshold parameter $K &gt; 0$ (i.e. the maximum carrying constant). Moreover, we present and justify nonstandard numerical methods constructed by specified balanced implicit methods
(BIMs). Their weak and $L^ p$-convergence follows from the fact that these methods with
local Lipschitz-continuous coefficients of logistic SDEs “produce” positive numerical approximations on bounded domain $[0, K]$ (a.s.). As commonly known, standard numerical
methods such as Taylor-type ones for SDEs fail to do that. Finally, asymptotic stability of
nontrivial equilibria $x_∗ = K$ is proven for both continuous time logistic SDEs and discrete
time approximations by BIMs. We exploit the technique of positive, sufficiently smooth
and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2411-m2023-0279},
url = {http://global-sci.org/intro/article_detail/jcm/24478.html}
}