679 research outputs found
Optimal projection filters with information geometry
We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo, Hanzon and Le Gland (1998, 1999) [19, 20], where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (2016) [6] where a direct L2 metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong, Brigo and Rossi Ferrucci (2019, 2018) [10, 9], based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges
Generally covariant state-dependent diffusion
Statistical invariance of Wiener increments under SO(n) rotations provides a
notion of gauge transformation of state-dependent Brownian motion. We show that
the stochastic dynamics of non gauge-invariant systems is not unambiguously
defined. They typically do not relax to equilibrium steady states even in the
absence of extenal forces. Assuming both coordinate covariance and gauge
invariance, we derive a second-order Langevin equation with state-dependent
diffusion matrix and vanishing environmental forces. It differs from previous
proposals but nevertheless entails the Einstein relation, a Maxwellian
conditional steady state for the velocities, and the equipartition theorem. The
over-damping limit leads to a stochastic differential equation in state space
that cannot be interpreted as a pure differential (Ito, Stratonovich or else).
At odds with the latter interpretations, the corresponding Fokker-Planck
equation admits an equilibrium steady state; a detailed comparison with other
theories of state-dependent diffusion is carried out. We propose this as a
theory of diffusion in a heat bath with varying temperature. Besides
equilibrium, a crucial experimental signature is the non-uniform steady spatial
distribution.Comment: 24 page
Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems
Let be a Riemannian -manifold. This paper is concerned with the
symmetry reduction of Brownian motion in and ramifications thereof in a
Hamiltonian context. Specializing to the case of polar actions we discuss
various versions of the stochastic Hamilton-Jacobi equation associated to the
symmetry reduction of Brownian motion and observe some similarities to the
Schr\"odinger equation of the quantum free particle reduction as described by
Feher and Pusztai. As an application we use this reduction scheme to derive
examples of quantum Calogero-Moser systems from a stochastic setting.Comment: V2 contains some improvements thanks to referees' suggestions; to
appear in Stochastics and Dynamic
Mean-Reverting Stochastic Processes, Evaluation of Forward Prices and Interest Rates
We consider mean-reverting stochastic processes and build self-consistent models for forward price dynamics and some applications in power industries. These models are built using the ideas and equations of stochastic differential geometry in order to close the system of equations for the forward prices and their volatility. Some analytical solutions are presented in the one factor case and for specific regular forward price/interest rates volatility. Those models will also play a role of initial conditions for a stochastic process describing forward price and interest rates volatility. Subsequently, the curved manifold of the internal space i.e. a discrete version of the bond term space (the space of bond maturing) is constructed. The dynamics of the point of this internal space that correspond to a portfolio of different bonds is studied. The analysis of the discount bond forward rate dynamics, for which we employed the Stratonovich approach, permitted us to calculate analytically the regular and the stochastic volatilities. We compare our results with those known from the literature.: Stochastic Differential Geometry, Mean-Reverting Stochastic Processes and Term Structure of Specific (Some) Economic/Finance Instruments
Symmetry and integrability for stochastic differential equations
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 (2010) \& (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. (2017)].Comment: to appear in J. Nonlin. Math.Phys. (2018
Random Lie-point symmetries of stochastic differential equations
We study the invariance of stochastic differential equations under random
diffeomorphisms, and establish the determining equations for random Lie-point
symmetries of stochastic differential equations, both in Ito and in
Stratonovich form. We also discuss relations with previous results in the
literature.Comment: In new version (November 2017) we have added an important ERRATUM.
Due to a trivial mistake in a formula, several examples should be revised;
more relevant, the qualitatove results of Section VIII turn out to be wrong,
as discussed in the erratum. See also arXiv:1711.0199
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