# Symmetry and integrability for stochastic differential equations

@article{Gaeta2017SymmetryAI, title={Symmetry and integrability for stochastic differential equations}, author={Giuseppe Gaeta and Claudia Lunini}, journal={Journal of Nonlinear Mathematical Physics}, year={2017}, volume={25}, pages={262 - 289} }

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) & 44 (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility of a system of SDEs to a lower dimensional one. We consider both “deterministic” symmetries and “random” ones, in the sense introduced recently by Gaeta and Spadaro [J. Math. Phys. 58 (2017)].

## 16 Citations

W-symmetries of Ito stochastic differential equations

- MathematicsJournal of Mathematical Physics
- 2019

We discuss W-symmetries of Ito stochastic differential equations, introduced in a recent paper by Gaeta and Spadaro [J. Math. Phys. 58, 053503 (2017)]. In particular, we discuss the general form of…

Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise

- Mathematics, Computer Science
- 2022

This work provides a classiﬁcation of scalar autonomous Ito stochastic dif-ferential equations with simple noise possessing symmetries, and extends previous classi ﬁcations in that it also considers recently introduced types of symmetry, in particular standard random symmetry, not considered in those.

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

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

Symmetry Analysis of the Stochastic Logistic Equation

- MathematicsSymmetry
- 2020

The general theory of symmetry of stochastic differential equations is used to obtain an explicit integration, i.e., an explicit formula for the process in terms of any single realization of the driving Wiener process.

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…

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…

Lie symmetry reductions and integrability of approximated small delay stochastic differential equations

- Mathematics
- 2020

This paper presents Lie symmetries of small delay stochastic differential equations (SDSDE). We derive an approximation of a small delay stochastic differential equation (SDSDE) equivalence to a…

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…

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…

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…

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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…

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Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. In order to…

On Lie-point symmetries for Ito stochastic differential equations

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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…

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