Symmetries and Functions of Markov Processes

Glover, Joseph; Mitro, Joanna B.

doi:10.1214/aop/1176990851

Cited by 22 publications

(23 citation statements)

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“…and Z i j are given in equation (42). SDEs (46) and (47) are in triangular form: indeed, the equation for R t depends only on R t , while the equation for Θ t is independent from Θ t itself.…”

Section: A Two Dimensional Examplementioning

confidence: 99%

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Albeverio¹,

Vecchi²,

Morando³

et al. 2020

Electron. J. Probab.

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Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markovian and non-Markovian examples of gauge and time symmetric processes are provided. The considered set of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes. Non trivial invariance results concerning a class of iterated random maps are obtained as special cases.

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Section: A Two Dimensional Examplementioning

confidence: 99%

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Albeverio¹,

Vecchi²,

Morando³

et al. 2020

Electron. J. Probab.

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“…There are two main approaches to this problem in the case of Brownian-motion-driven SDEs. The first approach, based on the Markovian property of solutions to a SDE, looks for the classical Lie symmetries of the Markov generator, which is an analytical object (see [1,9,15]). The second method, directly inspired by Lie ideas, consists in seeking for some semimartingale transformations leaving invariant the set of solutions to the considered SDE (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

A note on symmetries of diffusions within a martingale problem approach

2019

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A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.2 Preliminaries: second order geometry and Itô integration on manifoldIn this section, also in order to fix notations, we briefly recall some basic facts about second order geometry and Itô integration on manifolds. The interested reader is referred to [5,16,19] for proofs and further details. Second order geometryGiven a smooth manifold M , we denote by C ∞ (M ) the set of real-valued smooth functions defined on M . If F is a bundle with base manifold B, we denote by S(F ) the set of smooth sections of F . Finally, if M ′ is a manifold and n ∈ N, we denote by J n (M, M ′ ) the bundle of n jets of n times differentiable functions defined on M and taking values in M ′ . Let M be a smooth manifold and u be a global coordinate defined on R. The subset u −1 (0) ⊂ J 2 (M, R) is a submanifold of J 2 (M, R) and actually a vector subbundle of J 2 (M, R). Definition 2.1 The submanifold u −1 (0) ⊂ J 2 (M, R) is called the bundle of codiffusors of the manifold M and is denoted by τ * M .

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“…More generally Markov processes and their symmetries have been studied in a number of papers, see, e. g., [15,16,27,28,39]. Let us also mention that symmetry with respect to deterministic group of permutations has been exploited in de Finetti characterization of exchangeable processes, while processes which are symmetric with respect to groups of random transformations have been studied in [34] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Random transformations and invariance of semimartingales on Lie groups

Albeverio¹,

Vecchi²,

Morando³

et al. 2018

Preprint

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Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed.

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Symmetries and Functions of Markov Processes

Cited by 22 publications

References 0 publications

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

A note on symmetries of diffusions within a martingale problem approach

Random transformations and invariance of semimartingales on Lie groups

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