Integration of stochastic ordinary differential equations from a symmetry standpoint

We derive an algorithm for calculating Lie point symmetries of systems of stochastic ordinary differential equations (SODEs) of any order. From this algorithm, the following facts emerge. Symmetries of a SODE do not in general form a Lie algebra. The determining equations for Ito's equation are in general stochastic linear partial differential equations whereas for SODEs of order n ⩾2 the determining equations are linear deterministic partial differential equations that form an overdetemined system which is solvable by classical methods. For scalar second-order SODEs, we provide a complete classification of equations admitting finite-dimensional symmetry Lie algebras. This classification is applied to the integration of scalar second-order SODEs: in general a SODE admitting a two-dimensional symmetry algebra is not integrable by quadratures, although it is reducible to a homogeneous Ito equation. In particular, a scalar second-order SODE admitting a two-dimensional symmetry algebra with connected operators is linearizable. We also characterize integrable scalar second-order SODEs admitting three-dimensional symmetry algebras. Finally we show that a SODE can admit maximally a zero-, one-, two-, three- or four-dimensional Lie algebra.