Symmetries of stochastic differential equations using Girsanov transformations

2019/07/24 by Francesco C. De Vecchi, Paola Morando, Stefania Ugolini
Chemistry · Mathematics · Physics and Astronomy · #Applied mathematics #Class (philosophy) #Computer science #Girsanov theorem #Homogeneous space #Infinitesimal #Interpretation (philosophy) #Mathematical analysis #Mathematics #Measure (data warehouse) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Stochastic differential equation #Stochastic process #math-ph #math.MP #math.PR

paper · pdf · doi:10.1088/1751-8121/ab757d

published as 2020 J. Phys. A: Math. Theor. 53 135204

arxiv created 2019/07/24 · openalex publication_date 2020/02/12 · arxiv updated 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Abstract Aiming at enlarging the class of symmetries of a stochastic differential equation (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.

Citations

• The martingale property in the context of stochastic differential equations
• Reduction and reconstruction of stochastic differential equations via symmetries
• The geometry of differential constraints for a class of evolution PDEs
• A symmetry-adapted numerical scheme for SDEs
• A note on symmetries of diffusions within a martingale problem approach

Cited by

• Weak symmetries of stochastic differential equations driven by semimartingales with jumps