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This paper addresses existence and Ulam–Hyers stability (UHS) problems for an impulsive Caputo–Hadamard fractional neutral functional stochastic differential equation with infinite delay (FNFSDEwID). We first prove the existence and uniqueness of the solution using Banach fixed point theorem and standard stochastic analysis techniques. We then tackle the UHS under a Lipschitz condition on a bounded and closed interval. We end up with an illustrative example that corroborates our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the controllability for Hilfer fractional neutral stochastic differential equations with infinite delay and nonlocal conditions has been investigated. Using concepts from fractional calculus, semigroup of operators, fixed-point theory, measures of noncompactness, and stochastic theory the main controllability conclusion is attained. The applications of the key findings are finally illustrated with two examples. [ABSTRACT FROM AUTHOR]

We observe Poisson stable solutions for nonlinear stochastic functional differential equations (SFDEs) with finite delay. Firstly, we prove the existence and uniqueness of bounded (in square-mean sense) solutions and solution maps for SFDEs with finite delay by remoting start (or dissipative method) and classical pull-back attraction method. Then, based on the relationship between the solution and coefficients, we obtain such Poisson stable solutions by using Shcherbakov's comparability method in character of recurrence. Because the solutions of the delay equations are not-Markov, we employ the solution maps in the appropriate phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability. For illustration of our results, we give the application arising from stochastic Lotka–Volterra cooperative systems with distributed delay. • We prove the existence and uniqueness of L2-bounded (in square-mean sense) solution and solution map for nonlinear stochastic functional differential equations. The bounded solution and solution map are globally asymptotically stable in square-mean sense. • The L2-bounded solution possesses the same recurrent properties in distribution sense as the coefficients. • The L2-bounded solution map possesses the same recurrent properties in distribution sense as the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sufficient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G -Brownian motion (G -NSFDE). Moreover, the results extend the ones in the linear expectation case (Bai and Jiang, 2016) and nonlinear expectation framework (Huang and Yang, 2021). [ABSTRACT FROM AUTHOR]

Abstract: In this note we would like to give some remarks and open problems concerning comparison results for stochastic functional differential equations. Our starting point was the paper by [N. Halidias, Y. Ren, 2008. An existence theorem for stochastic functional differential equations with delays under weak assumptions, Statistics and Probability Letters, 78] where one can find some incorrect arguments in the comparison result. However, as we shall prove in this note, the results in that paper are true under some more restrictive conditions on the data. [Copyright &y& Elsevier]

In this work, we obtain some regularity results on stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded coefficient operators driven by white noise. We first establish some estimates on fundamental solutions which play an important role in the subsequent theory. Then we apply these estimates to some stochastic convolutions arising in stochastic delay differential equations to obtain the desired regularity property. [ABSTRACT FROM AUTHOR]