Symmetries of systems of stochastic differential equations with diffusion matrices of full rank

Abstract

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.

Symmetry of stochastic non-variational differential equations

“…In the stochastic case we obtain essentially the same result, but now it is convenient to consider separately the case of deterministic symmetries and that of random ones. Again the relevant results in this direction have been obtained by Kozlov [12,13] (see also [8,16]).…”

“…But it is also known that they have the same simple symmetries, and this both in the deterministic [24] and in the random [7] case. This fact is specially interesting, as the Kozlov theory [11,12,13] relating symmetry to integrability of SDEs only makes use of simple symmetries.…”

W-symmetries of Ito stochastic differential equations

“…We start with the case of simple deterministic symmetries. Here we have the following result, due to Kozlov [16] (see also [11]):…”

On Lie Group Classification of a Scalar Stochastic Differential Equation

“…Although a direct method can be used to obtain this Lie group classification, it is no longer practical for more complicated cases such as systems of SDEs [14] due to the complexity of the determining equations. Applications of symmetries for integration of scalar stochastic differential equations can be found in [13].…”