Ayman M. Mahmoud neutral stochastic differential equations equations first-order differential equations results 28 special case Lyapunov nontrivial solutions delay reliability trivial solution paper complement boundedness 2022-09-02T16:06 stochastic differential equations order system decisions differential equations third-order differential equation On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay solution function order differential equations complement article kind cases quadratic function stability stochastic boundedness behavior of solutions literature behavior nonlinear neutral stochastic differential equations 2022-04-07 This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x′(t)=y(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\prime }(t)=y(t)$\end{document}, y′(t)=z(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y^{\prime }(t) =z(t)$\end{document} the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature. https://doi.org/10.1186/s13662-022-03703-x effectiveness https://scigraph.springernature.com/explorer/license/ third order hypothesis articles article stochastic stability true 2022-04-07 criteria Ademola Adeleke T. 10.1186/s13662-022-03703-x doi Department of Mathematics, Obafemi Awolowo University, 220005, Ile-Ife, Nigeria Department of Mathematics, Obafemi Awolowo University, 220005, Ile-Ife, Nigeria 1687-1839 2731-4235 Advances in Continuous and Discrete Models Springer Nature 1 Department of Mathematics, Faculty of Science, New Valley University, 72511, El-Khargah, Egypt Department of Mathematics, Faculty of Science, New Valley University, 72511, El-Khargah, Egypt Mathematical Sciences dimensions_id pub.1146947930 Springer Nature - SN SciGraph project Statistics Applied Mathematics 2022