# Random Lie-point symmetries of stochastic differential equations

@article{Gaeta2017RandomLS, title={Random Lie-point symmetries of stochastic differential equations}, author={Giuseppe Gaeta and Francesco Spadaro}, journal={Journal of Mathematical Physics}, year={2017}, volume={58}, pages={053503} }

We study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random Lie-point symmetries of stochastic differential equations, both in Ito and in Stratonovich forms. We also discuss relations with previous results in the literature.

## 20 Citations

W-symmetries of backward stochastic differential equations, preservation of simple symmetries and Kozlov's theory

- MathematicsCommun. Nonlinear Sci. Numer. Simul.
- 2021

Random Lie symmetries of Itô stochastic differential equations

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2018

Lie point symmetries of stochastic differential equations (SDEs) which include transformations depending on the Brownian motion are considered. The corresponding Lie group transformations, acting in…

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

- MathematicsMathematics in Engineering
- 2022

We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several…

Recent advances in symmetry of stochastic differential equations.

- Mathematics
- 2018

We discuss some recent advances concerning the symmetry of stochastic differential equations, and in particular the interrelations between these and the integrability -- complete or partial -- of the…

Lie point symmetries of Stratonovich stochastic differential equations

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2018

This paper considers Lie point symmetries of stochastic differential equations (SDEs) in Stratonovich form. First, we derive the determining equations of the random symmetries, which correspond to…

Symmetry classification of scalar Ito equations with multiplicative noise

- MathematicsJournal of Nonlinear Mathematical Physics
- 2020

We provide a symmetry classification of scalar stochastic equations with multiplicative noise. These equations can be integrated by means of the Kozlov procedure, by passing to symmetry adapted…

Integration of the stochastic logistic equation via symmetry analysis

- MathematicsJournal of Nonlinear Mathematical Physics
- 2019

We apply the recently developed theory of symmetry of stochastic differential equations to a stochastic version of the logistic equation, obtaining an explicit integration, i.e. an explicit formula…

Symmetries of stochastic differential equations using Girsanov transformations

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

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…

Symmetry and integrability for stochastic differential equations

- Mathematics
- 2017

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A 43 (2010) &…

On Lie-point symmetries for Ito stochastic differential equations

- Mathematics
- 2017

In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make…

## References

SHOWING 1-10 OF 47 REFERENCES

Lie-point symmetries and stochastic differential equations

- Mathematics
- 1999

We discuss Lie-point symmetries of stochastic (ordinary) differential equations, and the interrelations between these and analogous symmetries of the associated Fokker-Planck equation for the…

Lie-point symmetries and stochastic differential equations: II

- Mathematics
- 2000

We complement the discussion of symmetries of Ito equations given in Gaeta and Rodriguez Quintero (1999 J. Phys. A: Math. Gen. 32 8485-505) by considering transformations acting on vector Wiener…

A Formal Approach for Handling Lie Point Symmetries of Scalar First-Order Itô Stochastic Ordinary Differential Equations

- Mathematics
- 2008

Abstract Many methods of deriving Lie point symmetries for Itô stochastic ordinary differential equations (SODEs) have surfaced. In the Itô calculus context both the formal and intuitive…

On maximal Lie point symmetry groups admitted by scalar stochastic differential equations

- Mathematics
- 2011

It is proved that the Lie point symmetry group admitted by a scalar stochastic differential equation (SDE) of order n ⩾ 3 is at most (n + 2) dimensional. This result supplements those for first- and…

Symmetries of first‐order stochastic ordinary differential equations revisited

- Mathematics
- 2007

Symmetries of stochastic ordinary differential equations (SODEs) are analysed. This work focuses on maintaining the properties of the Weiner processes after the application of infinitesimal…

On the definition of an admitted Lie group for stochastic differential equations with multi-Brownian motion

- Mathematics
- 2006

The definition of an admitted Lie group of transformations for stochastic differential equations has been already presented for equations with one-dimensional Brownian motion. The transformation of…

Conserved quantities and symmetries related to stochastic Hamiltonian systems

- Mathematics, Physics
- 2007

In this paper symmetries and conservation laws for stochastic dynamical systems are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratonovich…

A remark on symmetry of stochastic dynamical systems and their conserved quantities

- Mathematics, Physics
- 1995

The symmetry properties of stochastic dynamical systems described by a stochastic differential equation of Stratonovich type and related conserved quantities are discussed, extending previous results…

Symmetries of systems of stochastic differential equations with diffusion matrices of full rank

- Mathematics
- 2010

Lie point symmetries of a system of stochastic differential equations (SDEs) with diffusion matrices of full rank are considered. It is proved that the maximal dimension of a symmetry group admitted…