Logistic Stochastic Differential Equations with Power-Law (Journal of Computational Mathematics, 2025)
An analysis of logistic stochastic differential equations (SDEs) with general power-lawand 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 $K > 0$ (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 withlocal Lipschitz-continuous coefficients of logistic SDEs “produce” positive numerical approximations on bounded domain $[0, K]$ (a.s.). As commonly known, standard numericalmethods such as Taylor-type ones for SDEs fail to do that. Finally, asymptotic stability ofnontrivial equilibria $x_∗ = K$ is proven for both continuous time logistic SDEs and discretetime approximations by BIMs. We exploit the technique of positive, sufficiently smoothand Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.
Keywords: Logistic stochastic differential equations; Existence of bounded unique solutions; Asymptotic stability; positivity; Convergence; Balanced implicit methods.