3,057 research outputs found
Coarse-graining of overdamped Langevin dynamics via the Mori-Zwanzig formalism
The Mori–Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the importance of memory effects in determining the correct temporal behaviour of such systems
Information-geometric Markov Chain Monte Carlo methods using Diffusions
Recent work incorporating geometric ideas in Markov chain Monte Carlo is
reviewed in order to highlight these advances and their possible application in
a range of domains beyond Statistics. A full exposition of Markov chains and
their use in Monte Carlo simulation for Statistical inference and molecular
dynamics is provided, with particular emphasis on methods based on Langevin
diffusions. After this geometric concepts in Markov chain Monte Carlo are
introduced. A full derivation of the Langevin diffusion on a Riemannian
manifold is given, together with a discussion of appropriate Riemannian metric
choice for different problems. A survey of applications is provided, and some
open questions are discussed.Comment: 22 pages, 2 figure
Systemic Risk and Default Clustering for Large Financial Systems
As it is known in the finance risk and macroeconomics literature,
risk-sharing in large portfolios may increase the probability of creation of
default clusters and of systemic risk. We review recent developments on
mathematical and computational tools for the quantification of such phenomena.
Limiting analysis such as law of large numbers and central limit theorems allow
to approximate the distribution in large systems and study quantities such as
the loss distribution in large portfolios. Large deviations analysis allow us
to study the tail of the loss distribution and to identify pathways to default
clustering. Sensitivity analysis allows to understand the most likely ways in
which different effects, such as contagion and systematic risks, combine to
lead to large default rates. Such results could give useful insights into how
to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P.
Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer
Proceedings in Mathematics and Statistics, Vol. 110 2015
Exact propagation of open quantum systems in a system-reservoir context
A stochastic representation of the dynamics of open quantum systems, suitable
for non-perturbative system-reservoir interaction, non-Markovian effects and
arbitrarily driven systems is presented. It includes the case of driving on
timescales comparable to or shorter than the reservoir correlation time, a
notoriously difficult but relevant case in the context of quantum information
processing and quantum thermodynamics. A previous stochastic approach is
re-formulated for the case of finite reservoir correlation and response times,
resulting in a numerical simulation strategy exceeding previous ones by orders
of magnitude in efficiency. Although the approach is based on a memory
formalism, the dynamical equations propagated in the simulations are
time-local. This leaves a wide range of choices in selecting the system to be
studied and the numerical method used for propagation. For a series of tests,
the dynamics of the spin-boson system is computed in various settings including
strong external driving and Landau-Zener transitions.Comment: 7 pages, 4 figures. v2: inset in Fig. 2 and some text added, further
references. v3: minor correction
From rough path estimates to multilevel Monte Carlo
New classes of stochastic differential equations can now be studied using
rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this
paper we investigate, from a numerical analysis point of view, stochastic
differential equations driven by Gaussian noise in the aforementioned sense.
Our focus lies on numerical implementations, and more specifically on the
saving possible via multilevel methods. Our analysis relies on a subtle
combination of pathwise estimates, Gaussian concentration, and multilevel
ideas. Numerical examples are given which both illustrate and confirm our
findings.Comment: 34 page
Simple and extended Kalman filters : an application to term structures of commodity prices.
This article presents and compares two different Kalman filters. These methods provide a very interesting way to cope with the presence of non-observable variables, which is a frequent problem in finance. They are also very fast even in the presence of a large information volume. The first filter presented, which corresponds to the simplest version of a Kalman filter, can be used solely in the case of linear models. The second filter - the extended one - is a generalization of the first one, and it enables one to deal with non-linear models. However, it also introduces an approximation in the analysis, whose possible influence must be appreciated. The principles of the method and its advantages are first presented. It is then explained why it is interesting in the case of term structure models of commodity prices. Choosing a well-known term structure model, practical implementation problems are discussed and tested. Finally, in order to appreciate the impact of the approximation introduced for non-linear models, the two filters are compared.Term Structure; Commodity Future Prices; Kalman Filter;
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