Afrika Matematika https://doi.org/10.1007/s13370-020-00820-2 ORIGINAL ARTICLE Lie symmetry reductions and integrability of approximated small delay stochastic differential equations Aminu M. Nass 1 · Kassimu Mpungu 2 Received: 5 January 2020 / Accepted: 30 July 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020 Abstract This paper presents Lie symmetries of small delay stochastic differential equations (SDSDE). We derive an approximation of a small delay stochastic differential equation (SDSDE) equiv- alence to a non-delay stochastic differential equation, with an intent of constructing the corresponding Lie symmetry algebras. The method is applied to different examples of small delay stochastic differential equations to obtain symmetry algebras, reductions and solutions of the approximated non-delay stochastic differential equation. Keywords Lie symmetry · Small delay · Stochastic differential equation Mathematics Subject Classification 35B06 · 60H10 · 58J70 1 Introduction The theory of Lie symmetry method of differential equations initially developed by a Norwe- gian mathematician Sophus Lie and extended by Ovsiannikov [1], is one of the best methods used to reduce differential equations and establish their analytical solutions. Since it’s modern treatment, mathematicians have intensely studied and considerably improved the Lie sym- metry method of differential equations and its applications to differential models resulting from physical, financial and biological processes. We refer the reader to [1–9] for some of the most recent literature about the classical Lie symmetry theory, its extensions and applications to deterministic fractional and integer differential models resulting from physics, financial mathematics and population biology. Similarly, the theory of Lie group to non-deterministic differential equations has recently received significant attention. Thanks to Gaeta and Quintero [10], who extended the Lie sym- B Aminu M. Nass nssami001@myuct.ac.za Kassimu Mpungu kmpungu@uhb.edu.sa 1 Department of Actuarial Science, Federal University Dutse, P.M.B 7156, Dutse, Jigawa State, Nigeria 2 Department of Mathematics, University of Hafr Al-Batin, P.O. Box 1803, Hafr Al-Batin, Saudi Arabia 123