1,305 research outputs found
An extension of Wiener integration with the use of operator theory
With the use of tensor product of Hilbert space, and a diagonalization
procedure from operator theory, we derive an approximation formula for a
general class of stochastic integrals. Further we establish a generalized
Fourier expansion for these stochastic integrals. In our extension, we
circumvent some of the limitations of the more widely used stochastic integral
due to Wiener and Ito, i.e., stochastic integration with respect to Brownian
motion. Finally we discuss the connection between the two approaches, as well
as a priori estimates and applications.Comment: 13 page
Modulation spaces, Wiener amalgam spaces, and Brownian motions
We study the local-in-time regularity of the Brownian motion with respect to
localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces
W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time
to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the
indices is optimal. Moreover, with the Wiener measure \mu on T, we show that
(M^{p, q}_s (T), \mu) and (W^{p, q}_s (T), \mu) form abstract Wiener spaces for
the same range of indices, yielding large deviation estimates. We also
establish the endpoint regularity of the periodic Brownian motion with respect
to a Besov-type space \ft{b}^s_{p, \infty} (T). Specifically, we prove that the
Brownian motion belongs to \ft{b}^s_{p, \infty} (T) for (s-1) p = -1, and it
obeys a large deviation estimate. Finally, we revisit the regularity of
Brownian motion on usual local Besov spaces B_{p, q}^s, and indicate the
endpoint large deviation estimates.Comment: 35 pages. The introduction is expanded. Appendices are added (A:
derivation of Fourier-Wiener series, B: passing estimates from T to bounded
intervals on R.) To appear in Adv. Mat
Asymptotic equivalence for inference on the volatility from noisy observations
We consider discrete-time observations of a continuous martingale under
measurement error. This serves as a fundamental model for high-frequency data
in finance, where an efficient price process is observed under microstructure
noise. It is shown that this nonparametric model is in Le Cam's sense
asymptotically equivalent to a Gaussian shift experiment in terms of the square
root of the volatility function and a nonstandard noise level. As an
application, new rate-optimal estimators of the volatility function and simple
efficient estimators of the integrated volatility are constructed.Comment: Published in at http://dx.doi.org/10.1214/10-AOS855 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Bayesian Numerical Homogenization
Numerical homogenization, i.e. the finite-dimensional approximation of
solution spaces of PDEs with arbitrary rough coefficients, requires the
identification of accurate basis elements. These basis elements are oftentimes
found after a laborious process of scientific investigation and plain
guesswork. Can this identification problem be facilitated? Is there a general
recipe/decision framework for guiding the design of basis elements? We suggest
that the answer to the above questions could be positive based on the
reformulation of numerical homogenization as a Bayesian Inference problem in
which a given PDE with rough coefficients (or multi-scale operator) is excited
with noise (random right hand side/source term) and one tries to estimate the
value of the solution at a given point based on a finite number of
observations. We apply this reformulation to the identification of bases for
the numerical homogenization of arbitrary integro-differential equations and
show that these bases have optimal recovery properties. In particular we show
how Rough Polyharmonic Splines can be re-discovered as the optimal solution of
a Gaussian filtering problem.Comment: 22 pages. To appear in SIAM Multiscale Modeling and Simulatio
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains
Metaplex networks: influence of the exo-endo structure of complex systems on diffusion
In a complex system the interplay between the internal structure of its
entities and their interconnection may play a fundamental role in the global
functioning of the system. Here, we define the concept of metaplex, which
describes such trade-off between internal structure of entities and their
interconnections. We then define a dynamical system on a metaplex and study
diffusive processes on them. We provide analytical and computational evidences
about the role played by the size of the nodes, the location of the internal
coupling areas, and the strength and range of the coupling between the nodes on
the global dynamics of metaplexes. Finally, we extend our analysis to two
real-world metaplexes: a landscape and a brain metaplex. We corroborate that
the internal structure of the nodes in a metaplex may dominate the global
dynamics (brain metaplex) or play a regulatory role (landscape metaplex) to the
influence of the interconnection between nodes.Comment: 28 pages, 19 figure
Modelling fluctuations of financial time series: from cascade process to stochastic volatility model
In this paper, we provide a simple, ``generic'' interpretation of
multifractal scaling laws and multiplicative cascade process paradigms in terms
of volatility correlations. We show that in this context 1/f power spectra, as
observed recently by Bonanno et al., naturally emerge. We then propose a simple
solvable ``stochastic volatility'' model for return fluctuations. This model is
able to reproduce most of recent empirical findings concerning financial time
series: no correlation between price variations, long-range volatility
correlations and multifractal statistics. Moreover, its extension to a
multivariate context, in order to model portfolio behavior, is very natural.
Comparisons to real data and other models proposed elsewhere are provided.Comment: 21 pages, 5 figure
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