77 research outputs found
Robust filtering: Correlated noise and multidimensional observation
In the late seventies, Clark [In Communication Systems and Random Process
Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734,
Sijthoff & Noordhoff] pointed out that it would be natural for , the
solution of the stochastic filtering problem, to depend continuously on the
observed data . Indeed, if the signal and the observation
noise are independent one can show that, for any suitably chosen test function
, there exists a continuous map , defined on the space of
continuous paths endowed with the uniform convergence
topology such that , almost surely; see, for example,
Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO
Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff],
Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z.
Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27
(1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and
Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic
Systems: The Mathematics of Filtering and Identification and Applications,
Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The
Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this
type of robust representation is also possible when the signal and the
observation noise are correlated, provided the observation process is scalar.
For a general correlated noise and multidimensional observations such a
representation does not exist. By using the theory of rough paths we provide a
solution to this deficiency: the observation process is "lifted" to the
process that consists of and its corresponding L\'{e}vy area
process, and we show that there exists a continuous map , defined
on a suitably chosen space of H\"{o}lder continuous paths such that
, almost surely.Comment: Published in at http://dx.doi.org/10.1214/12-AAP896 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Examples of renormalized SDEs:SPDERF 2016: Stochastic Partial Differential Equations and Related Fields
We demonstrate two examples of stochastic processes whose lifts to geometric
rough paths require a renormalisation procedure to obtain convergence in rough
path topologies. Our first example involves a physical Brownian motion subject
to a magnetic force which dominates over the friction forces in the small mass
limit. Our second example involves a lead-lag process of discretised fractional
Brownian motion with Hurst parameter , in which the stochastic
area captures the quadratic variation of the process. In both examples, a
renormalisation of the second iterated integral is needed to ensure convergence
of the processes, and we comment on how this procedure mimics negative
renormalisation arising in the study of singular SPDEs and regularity
structures.Comment: 13 page
From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index
{Let be a -dimensional fractional Brownian motion
with Hurst index , or more generally a Gaussian process whose paths
have the same local regularity. Defining properly iterated integrals of is
a difficult task because of the low H\"older regularity index of its paths. Yet
rough path theory shows it is the key to the construction of a stochastic
calculus with respect to , or to solving differential equations driven by
.
We intend to show in a series of papers how to desingularize iterated
integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure
defined by a limit in law procedure. Convergence is proved by using "standard"
tools of constructive field theory, in particular cluster expansions and
renormalization. These powerful tools allow optimal estimates, and call for an
extension of Gaussian tools such as for instance the Malliavin calculus.
After a first introductory paper \cite{MagUnt1}, this one concentrates on the
details of the constructive proof of convergence for second-order iterated
integrals, also known as L\'evy area
Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems
In recent years, substantial progress was made towards understanding
convergence of fast-slow deterministic systems to stochastic differential
equations. In contrast to more classical approaches, the assumptions on the
fast flow are very mild. We survey the origins of this theory and then revisit
and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1
(2016), 479-520], taking into account recent progress in -variation and
c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the
Conference in Honor of the 75th Birthday of S.R.S. Varadha
Stochastic many-particle model for LFP electrodes
In the framework of non-equilibrium thermodynamics, we derive a new model for many-particle electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by both different particle sizes and surface fluctuations leading to a system of stochastic differential equations. An explicit relation between battery voltage and current controlled by the thermodynamic state variables is derived. This voltage–current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate-limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate-limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltage–charge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates
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