Symmetries of Stochastic Differential Equations using Girsanov transformations

Abstract

Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models.

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Cited by in corpus (8)

• Weak symmetries of stochastic differential equations driven by semimartingales with jumps
• Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries
• Noether theorem in stochastic optimal control problems via contact symmetries
• White Noise Space Analysis and Multiplicative Change of Measures
• Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise
• 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