Lie point group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. The admitted symmetry group can be zero, one, two or three dimensional. If an equation admits a three-dimensional symmetry group, it can be transformed into the equation of Brownian motion by a change of variables with non-random time transformation. Such equations can be integrated by quadratures. There are described drift coefficients for which SDEs with constant diffusion coefficients admit two-and three-dimensional symmetry groups.

Lie point symmetries of a system of stochastic differential equations (SDEs) with diffusion matrices of full rank are considered. It is proved that the maximal dimension of a symmetry group admitted by a system of n SDEs is n + 2. In addition, such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality. These results are applied to compute the Lie group classification of a system of two SDEs. The classification is obtained with the help of non-equivalent realizations of real Lie algebras by fiber-preserving vector fields in 1 + 2 variables. Possibilities of using symmetries for integration of SDEs by quadratures are discussed.

This article is part of a research program the aim of which is to turn Lie group theory into an efficient tool for classifying, transforming and solving a further class of equations, namely Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable x and one dependent variable y. As opposed to ODEs the variable x figures in more than one point (we consider the case of two points, x and x − ). The dependent variable y and its derivatives figure in both x and x − . This is not a discretization since both x and y remain continuous. Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension n with 0 ≤ n ≤ 6 for genuinely nonlinear DODSs and n = ∞ for linear or linearizable ones. The symmetry algebras can be used to perform symmetry reduction and obtain exact particular (invariant) solutions.

One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two-and three-dimensional groups of Lagrangian symmetries are considered.

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.