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

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.