Integration of stochastic ordinary differential equations from a symmetry standpoint (Q2716823)

The authors extend Lie's classical symmetry algorithm for deterministic differential equations to stochastic ordinary differential equations (SODE). They characterise the (Lie point) symmetries of SODEs of order \(n\), that is the generators of a 1-parameter semigroup of transformations on \(\mathbb{R} \times \mathbb{R}^n\) (time-state space) which leave the SODE invariant: A vectorfield on \(\mathbb{R} \times \mathbb{R}^n\) is a symmetry, iff it satisfies a certain stochastic differential equation. (Generally, the symmetries of a SODE do not form a Lie algebra.) The characterization of a symmetry is then used to classify all scalar 2nd order SODEs which admit a Lie algebra of symmetries of finite dimension (which turns out to be maximally 4) and study whether or not these SODEs are linearizable, homogenizable or integrable by quadratures.