1,387 research outputs found
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 -prolongations of vector
fields, and hence of -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 , and we speak of -prolongations of vector fields
and -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
- …