Abstract
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.
Highlights
The study of symmetries and invariance properties of ordinary and partial differential equations (ODEs and PDEs, respectively) is a classical and well-developed research field and provides a powerful tool for both computing some explicit solutions to the equations and analyzing their qualitative behavior.The study of invariance properties of finite or infinite dimensional stochastic differential equations (SDEs) is, in comparison, less developed and a systematic study could be fruitful from both the practical and the theoretical point of view.The knowledge of some closed formulas is important in many applications of stochastic processes since it permits to develop faster and cheaper numerical algorithms for theWeak symmetries of SDEs simulation of the process or to evaluate interesting quantities related to it
The use of closed formulas allows the application of simpler statistical methods for the calibration of models
The presence of symmetries and invariance properties is a strong clue for the possibility of closed formulas
Summary
The study of symmetries and invariance properties of ordinary and partial differential equations (ODEs and PDEs, respectively) is a classical and well-developed research field (see [10, 51]) and provides a powerful tool for both computing some explicit solutions to the equations and analyzing their qualitative behavior. The second problem is addressed by using the new notions of invariance of a semimartingale defined on a Lie group introduced in [1] These two notions are extensions of predictable transformations which preserve the law of n dimensional Brownian motion and α-stable processes studied for example in [39, Chapter 4]. Given an SDE Ψ and a driving process Z with gauge symmetry group Ξg and time symmetry Γr, we are able to define a stochastic transformation T = (Φ, B, η), where Φ, η are a diffeomorphism and a density of a time change as in the Brownian setting, while B is a function taking values in G (in the Brownian setting G is the group of rotation in Rn). The second novelty of the paper is given by our explicit approach: we provide many results which permit to check explicitly whether a semimartingale admits given gauge and time symmetries and to compute stochastic transformations which are symmetries of a given SDE.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
