Symmetries of stochastic differential equations using Girsanov transformations

Aiming at enlarging the class of symmetries of an 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

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Stochastic symmetries and related invariance properties of \ufb01nite dimensional SDEs driven by general c`adl`ag semimartingales taking values in Lie groups are de\ufb01ned and investigated. The considered set of SDEs, \ufb01rst introduced by S. Cohen, includes a\ufb03ne and Marcus type SDEs as well as smooth SDEs driven by L\ub4evy 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 a\ufb03ne iterated random maps as well as (weak) symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs

On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

Local and nonlocal solvable structures in ODEs reduction

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure

Variational principles for involutive systems of vector fields

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of an higher order variational principle, and how this extends to involutive systems of vector fields.Comment: 31 pages. To appear in International Journal of Geometric Methods in Modern Physics (IJGMMP

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

A note on symmetries of diffusions within a martingale problem approach

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and It\uf4 integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes