Stochastic Wiener Filter in the White Noise Space

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this spaces in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions

Functorial Statistical Physics: Feynman--Kac Formulae and Information Geometries

The main results of this paper comprise proofs of the following two related facts: (i) the Feynman--Kac formula is a functor $F_*$, namely, between a stochastic differential equation and a dynamical system on a statistical manifold, and (ii) a statistical manifold is a sheaf generated by this functor with a canonical gluing condition. Using a particular locality property for $F_*$, recognised from functorial quantum field theory as a `sewing law,' we then extend our results to the Chapman--Kolmogorov equation {\it via} a time-dependent generalisation of the principle of maximum entropy. This yields a partial formalisation of a variational principle which takes us beyond Feynman--Kac measures driven by Wiener laws. Our construction offers a robust glimpse at a deeper theory which we argue re-imagines time-dependent statistical physics and information geometry alike.Comment: 8+1 pages. Announcemen

Self-Excited Multifractal Dynamics

We introduce the self-excited multifractal (SEMF) model, defined such that the amplitudes of the increments of the process are expressed as exponentials of a long memory of past increments. The principal novel feature of the model lies in the self-excitation mechanism combined with exponential nonlinearity, i.e. the explicit dependence of future values of the process on past ones. The self- excitation captures the microscopic origin of the emergent endogenous self-organization properties, such as the energy cascade in turbulent flows, the triggering of aftershocks by previous earthquakes and the "reflexive" interactions of financial markets. The SEMF process has all the standard stylized facts found in financial time series, which are robust to the specification of the parameters and the shape of the memory kernel: multifractality, heavy tails of the distribution of increments with intermediate asymptotics, zero correlation of the signed increments and long-range correlation of the squared increments, the asymmetry (called "leverage" effect) of the correlation between increments and absolute value of the increments and statistical asymmetry under time reversal