Geodesic
Equations for covariance functions of the state vector of a linear system of stochastic differential equations with finite concentrated and distributed delays
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 46-55

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In this paper, we present a step-by-step method for the approximate analytical calculation of the matrix of covariance functions for a system of linear stochastic ordinary integro-differential equations with finite concentrated and distributed delays perturbed by additive fluctuations in the form of a vector standard Wiener process with independent components. The method proposed is a combination of the classical method of steps and the expansion of the state space and consists of several stages that make it possible to pass from a non-Markov system of stochastic equations to a chain of Markov systems without delay. Based on these systems, we construct sequences of systems of auxiliary linear ordinary differential equations for elements of vectors of mathematical expectations and covariance matrices of extended state vectors, and then obtaib the required equations for covariance functions.
Keywords: state vector, covariance function, stochastic integro-differential equation, concentrated delay, distributed delay, step-by-step method 
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     author = {I. E. Poloskov},
     title = {Equations for covariance functions of the state vector of a linear system of stochastic differential equations with finite concentrated and distributed delays},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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     publisher = {mathdoc},
     volume = {233},
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I. E. Poloskov. Equations for covariance functions of the state vector of a linear system of stochastic differential equations with finite concentrated and distributed delays. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 46-55. http://geodesic.mathdoc.fr/item/INTO_2024_233_a4/