Logistic Stochastic Differential Equations with Power-Law

Abstract

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 $K > 0$ (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.