Some Recent Developments on Lie Symmetry Analysis of Stochastic Differential Equations

We present a brief review of recent progresses on Lie symmetry analysis of stochastic differential equations (SDEs). In particular, we consider some general definitions of symmetries for Brownian motion driven SDEs, as well as of weak and gauge symmetries of SDEs driven by discrete-time semimartingales. Some applications of Lie symmetry analysis to reduction and reconstruction of SDEs, Kolmogorov equation and numerical schemes for SDEs are discussed. Studies on random symmetries of SDEs, as well as extension of Noether theorem on invariants to stochastic systems and the finding of finite-dimensional solutions to SPDEs are also briefly reviewed.