Symmetries of stochastic differential equations: A geometric approach

Abstract

A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.

Symmetry of stochastic non-variational differential equations

“…Note that his approach is entirely through the associated diffusion (hence deterministic) equation. The interested reader is referred to his paper [73] (announced in [74]).…”

Symmetries of stochastic differential equations using Girsanov transformations

“…The application of Lie symmetry analysis techniques to stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) has received a growing interest in recent years. Without claiming to be exhaustive, from the first studies on Brownianmotion-driven SDEs (see [3,7,24]) and Markov processes (see [8,26,27,36]) many different notions of symmetries for Brownian-motion-driven SDEs has been proposed (see the symmetries with random time change [20,45,47], W-symmetry in [21,22], weak symmetries in [15], random symmetries in [23,25,32] and symmetries of diffusions in [16]). Moreover, random transformations of semimartingales on Lie groups has been introduced (see [1]) and the notion of symmetry has been extended also to SDEs driven by general semimartingales (see [2,35]).…”

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

“…We will consider in particular the approach which parallels the usual treatment of deterministic equations in the stochastic case (to which we gave several contributions in recent years); for a discussion of other approaches, including earlier attempts, see e.g., the review paper [22]. See also [66,67] for an approach relating symmetries to Girsanov theory, and more generally [68,69,70,71,72] for a related concurrent approach.…”