TY - JOUR
T1 - Logistic Stochastic Differential Equations with Power-Law
AU - Schurz , Henri
JO - Journal of Computational Mathematics
VL - 5
SP - 1219
EP - 1237
PY - 2025
DA - 2025/09
SN - 43
DO - http://doi.org/10.4208/jcm.2411-m2023-0279
UR - https://global-sci.org/intro/article_detail/jcm/24478.html
KW - Logistic stochastic differential equations, Existence of bounded unique solutions, Asymptotic stability, positivity, Convergence, Balanced implicit methods.
AB - <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>