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.

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• Weak symmetries of stochastic differential equations driven by semimartingales with jumps
• Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries
• A symmetry-adapted numerical scheme for SDEs
• Symmetries and invariance properties of stochastic differential equations driven by semimartingales with jumps
• Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field
• On the integration of Ito equations with a random or a W-symmetry
• Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations