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In this survey, we show that various stochastic optimization problems arising in
option theory, in dynamical allocation problems, and in the microeconomic theory
of intertemporal consumption choice can all be reduced to the same problem of
representing a given stochastic process in terms of running maxima of another
process. We describe recent results of Bank and El Karoui (2002) on the general
stochastic representation problem, derive results in closed form for Lévy processes
and diffusions, present an algorithm for explicit computations, and discuss some
applications.
This paper is concerned with the effficient implementation of transparent boundary conditions
(TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry.
In [1] a discrete TBC of convolution type was derived from the fully discretized whole?space
problem that is reflection?free and yields an unconditionally stable scheme. Since the discrete
TBC includes a convolution with respect to range with a weakly decaying kernel, its
numerical evaluation becomes very costly for long-range simulations.
As a remedy we construct new approximative transparent boundary conditions involving
exponential sums as an approximation to the convolution kernel. This special approximation
enables us to use a fast evaluation of the convolution type boundary condition.
This new approach was outlined in detail in [2] for the standard "parabolic" equation.
Differential algebraic equations with properly stated leading term are equations of the form A(x(t),t)(d(x(t),t))'+b(x(t),t)=0 with in some sense well-matched coefficients. Systems resulting from the modified nodal analysis (MNA) in circuit simulation promptly fit into this form. Recent results concerning solvability and numerical treatment of those equations are discussed. An index notion that works via linearization is given. This allows for index criteria just in terms of the coefficients A,d,b and their first partial derivatives, no further derivative arrays are used.
We present a way to efficiently treat the well-known transparent boundary
conditions for the Schrödinger equation. Our approach is based on two ideas:
firstly, to derive a discrete transparent boundary condition (DTBC) based on the
Crank-Nicolson finite difference scheme for the governing equation. And, secondly,
to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for
a rapid recursive calculation of the convolution. We illustrate the efficiency of the
proposed method on several examples.
The index of DAE systems arising from linear quadratic optimal control problems is considered. Necessary and sufficient conditions ensuring regularity with tractability index one are proved. Then, it is shown that if the control problem DAE is regular with index one and if the leading term of the DAE to be controlled is given by one full-column-rank and one full-row-rank matrix, then it has a Hamiltonian inherent explicit ODE. For problems with regular index zero or index one DAEs to be controlled, the DAE of the control problem is shown to be regular with tractability index one or three, depending on whether the control coefficient R is singular.
By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint equation. When applied to boundary value problems, the result provides a constructive criterion for unique solvability and allows reducing the problem to initial value problems and linear algebraic equations.
We consider a particle constrained to a submanifold ? of the
configuration space Rm. Using that the notion of holonomic constraints coincides
with integrability of the corresponding vector field, we show how this property
naturally determines local coordinates on ?. We give a rigorous justification for
the calculation of the mean force along a constrained coordinate, and we provide
a concise geometrical interpretation of the different contributions to the mean
force in terms of the unconstrained vector field and extrinsic curvature properties
of ? in Rm. Our approach gives rise to a Hybrid Monte-Carlo based algorithm
that can be used to compute the mean force acting on selected coordinates in the
context of thermodynamic free energy statistics.
We analyze an interactive model of credit ratings where external shocks, initially
affecting only a small number of firms, spread by a contagious chain reaction to the
entire economy. Counterparty relationships along with discrete adjustments of credit
ratings generate a transition mechanism that allows the financial distress of one firm
to spill over to its business partners. Such a contagious infectious of financial distress
constitutes a source of intrinsic risk for large portfolios of credit sensitive securities that
cannot be “diversified away.” We provide a complete characterization of the fluctuations
of credit ratings in large economies when adjustments follow a threshold rule. We also
analyze the effects of downgrading cascades on aggregate losses of credit portfolios. We
show that the loss distribution has a power-law tail if the interaction between different
companies is strong enough.
Stability of Linear Stochastic Difference Equations in Strategically Controlled Random Environments
(2004)
We consider the stochastic sequence fYtgt2N defined recursively by the linear relation
Yt+1 = AtYt+Bt in a random environment. The environment is described by the stochastic
process f(At;Bt)gt2N and is under the simultaneous control of several agents playing a
discounted stochastic game. We formulate sufficient conditions on the game which ensure
the existence of Nash equilibrium in Markov strategies which has the additional property
that, in equilibrium, the process fYtgt2N converges in distribution to a stationary regime.
We study the effect of investor inertia on stock price fluctuations with a market microstructure
model comprising many small investors who are inactive most of the time.
It turns out that semi-Markov processes are tailor made for modelling inert investors.
With a suitable scaling, we show that when the price is driven by the market imbalance,
the log price process is approximated by a process with long range dependence
and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently,
investor inertia may lead to arbitrage opportunities for sophisticated market
participants. The mathematical contributions are a functional central limit theorem for
stationary semi-Markov processes, and approximation results for stochastic integrals
of continuous semimartingales with respect to fractional Brownian motion.
We consider a financial market model with a large number of interacting
agents. Investors are heterogeneous in their expectations
about the future evolution of an asset price process. Their current
expectation is based on the previous states of their “neighbors” and
on a random signal about the “mood of the market.” We analyze the
asymptotics of both aggregate behavior and asset prices. We give sufficient
conditions for the distribution of equilibrium prices to converge to
a unique equilibrium, and provide a microeconomic foundation for the
use of diffusion models in the analysis of financial price fluctuations.
We consider general economies in which rational agents interact locally. The local aspect
of the interactions is designed to represent in a simple abstract way social interactions, that
is, socioeconomic environments in which markets do not mediate all of agents' choices, and
each agent's choice might be in part determined, for instance, by family, peer group, or ethnic
group effects. We study static as well as dynamic infinite horizon economies; we allow for
economies with incomplete information, and we consider jointly global and local interactions,
to integrate e.g., global externalities and markets with peer and group effects. We provide
conditions under which such economies have rational expectations equilibria.
We illustrate the effects of local interactions when agents are rational by studying in detail
the equilibrium properties of a simple economy with quadratic preferences which captures, in
turn, local preferences for conformity, habit persistence, and preferences for status or adherence
to aggregate norms of behavior.
We give sufficient conditions for a non-zero sum discounted stochastic game with
compact and convex action spaces and with norm-continuous transition probabilities,
but with possibly unbounded state space, to have a Nash equilibrium in homogeneous
Markov strategies that depends in a Lipschitz continuous manner on the current state. If
the underlying state space is compact this yields the existence of a stationary equilibrium.
Stochastic games with weakly interacting players provide a probabilistic framework within
which to study strategic behavior in models of non-market interactions.
This paper addresses the regularization of pointwise state constraints in optimal
control problems. By analyzing the associated dual problem, it is shown that the regularized problems
admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of
the regularized problem converges to the one of the original state constrained problem. The results
of our analysis are confirmed by numerical tests.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
We provide a number of new construction techniques for cubical complexes and cubical
polytopes, and thus for cubifications (hexahedral mesh generation). As an application we
obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein
bottle). This confirms an existence conjecture of Hetyei (1995).
More systematically, we prove that every normal crossing codimension one immersion of
a compact 2-manifold into R3 is PL-equivalent to a dual manifold immersion of a cubical
4-polytope. As an instance we obtain a cubical 4-polytope with a cubation of Boy's surface
as a dual manifold immersion, and with an odd number of facets. Our explicit example has
17 718 vertices and 16 533 facets. Thus we get a parity changing operation for 3-dimensional
cubical complexes (hexa meshes); this solves problems of Eppstein, Thurston, and others.
We discuss solvers for Sylvester, Lyapunov, and Stein equations
that are available in the SLICOT Library (Subroutine
Library In COntrol Theory). These solvers offer improved
efficiency, reliability, and functionality compared to corresponding
solvers in other computer-aided control system design
packages. The performance of the SLICOT solvers is
compared with the corresponding MATLAB solvers.
The paper presents a unified approach to local likelihood estimation
for a broad class of nonparametric models, including e.g. the regression,
density, Poisson and binary response model. The method extends
the adaptive weights smoothing (AWS) procedure introduced in Polzehl
and Spokoiny (2000) in context of image denoising. Performance of the
proposed procedure is illustrated by a number of numerical examples
and applications to density or volatility estimation, classification and
estimation of the tail index parameter. We also establish a number of
important theoretical results on properties of the proposed procedure.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.
Error estimates for the numerical approximation of boundary semilinear elliptic control problems
(2004)
We study the numerical approximation of boundary optimal control problems governed
by semilinear elliptic partial differential equations with pointwise constraints on the control.
The analysis of the approximate control problems is carried out. The uniform convergence of discretized
controls to optimal controls is proven under natural assumptions by taking piecewise constant
controls. Finally, error estimates are established.
Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints
(2004)
A class of quadratic optimization problems in Hilbert spaces is considered, where
pointwise box constraints and constraints of bottleneck type are given. The main focus is to prove the
existence of regular Lagrange multipliers in L2-spaces. This question is solved by investigating the
solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control
problems for elliptic and parabolic partial differential equations with mixed pointwise control-state
constraints.
Let Ex be a collection of i.i.d. exponential random
variables. Symmetric Bouchaud’s model on Z2 is a Markov chain
X(t) whose transition rates are given by wxy = ν exp(−βEx ) if x,
y are neighbours in Z2 . We study the behaviour of two correlation functions: P[X(tw + t) = X(tw )] and P X(t ) = X(tw )∀t ∈
[tw , tw + t] . We prove the (sub)aging behaviour of these functions
when β > 1.
A popular model to describe credit risk in practice is CreditRisk
+
and in
this paper a Fourier inversion to obtain the distribution of the credit loss is
proposed. A deeper analysis of the Fourier transformation showed that there
are at least two methods to obtain the distribution although the corresponding
characteristic function is not integrable.
The CreditRisk
+
model will be extended such, that general dependent sec-
tor variables can be taken into consideration, for example dependent lognormal
sector variables. Then the transfer to a continuous time model will be per-
formed and the sector variables become processes, more precisely geometric
Brownian motions.
To have a time continuous credit risk model is an important step to combine
this model with market risk. Additionally a portfolio model will be presented
where the changes of the spreads are driven by the sector variables. Using a
linear expansion of the market risk, the distribution of this portfolio can be
determined. In the special case that there is no credit risk, this model yields
the well known Delta normal approach for market risk, hence a link between
credit risk and market risk has been established.
The CreditRisk model launched by CSFB in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit
risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk, equivalent to an algorithm recently proposed by
Giese, based on well-known expansions of the logarithm and the exponential
of a power series. We show that it is advantageous to the Panjer recursion
advocated in the original CreditRisk
document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the
computational complexity of the resulting algorithm is stated.
We introduce a new Monte Carlo method for constructing the exercise
boundary of an American option in a generalized Black-Scholes framework.
Based on a known exercise boundary, it is shown how to price and hedge the
American option by Monte Carlo simulation of suitable probabilistic represen-
tations in connection with the respective parabolic boundary value problem.
The methods presented are supported by numerical simulation experiments.
In this paper we investigate the use of parallel computing to deal with the high computational cost of numerical algorithms for model reduction of large linear descriptor systems. The state-space truncation methods considered here are composed of iterative schemes which can be efficiently implemented on parallel architectures using existing parallel linear algebra libraries. Our experimental results on a cluster of Intel Pentium processors show the performance of the parallel algorithms.
We discuss a parallel library of efficient algorithms for model reduction of largescale
systems with state-space dimension up to O(104). We survey the numerical
algorithms underlying the implementation of the chosen model reduction methods.
The approach considered here is based on state-space truncation of the system
matrices and includes absolute and relative error methods for both stable and unstable
systems. In contrast to serial implementations of these methods, we employ
Newton-type iterative algorithms for the solution of the major computational tasks.
Experimental results report the numerical accuracy and the parallel performance of
our approach on a cluster of Intel Pentium II processors.
We describe a prototype web service for model reduction of very large-scale linear systems, with
dimension in the order of millions of states, that includes a user-friendly interface designed so that the computation
can be easily performed via the HTTP protocol. Access via a web browser isolates the user of the service from the
complexities of installing and using the parallel model reduction codes and the maintenance of the hardware. In case
the routines are found to be appropriate for the problem the user needs to solve, the library can be then downloaded
and installed on the user’s own computing resources.
This paper illustrates the major issues of the access procedure by means of graphical examples, and describes
the structure and implementation of the remote model reduction service. The service is offered in a cluster of Linux
machines.
A Structure-Preserving Method for Generalized Algebraic RiccatiEquations Based on Pencil Arithmetic
(2004)
This paper describes a numerical method for extracting the stable
right deflating subspace of a matrix pencil Z Y using
a spectral projection method. It has several advantages compared
to other spectral projection methods like the sign function
method. In particular it avoids the rounding error induced
loss of accuracy associated with matrix inversions. The new algorithm
is particularly well adapted to solving continuous-time
algebraic Riccati equations. In numerical examples, it solves
Riccati equations to high accuracy.
We study the optimization of three dimensional curved rods and of shells
under minimal regularity assumptions for the geometry. The results that we
establish concern the existence of optimal shapes and the sensitivity analysis.
We also compute several numerical examples for the curved rods. The models
that we use have been investigated in our previous work [11], [16] and a
complete study of the Kirchhoff-Love arches and their optimization has been
performed in [10].
We prove new properties for the linear isotropic elasticity system and for
thickness minimization problems. We also present very recent results concerning
shape optimization problems for three-dimensional curved rods and
for shells. The questions discussed in this paper are related to the control
variational method and to control into coefficients problems.
In this paper a nonlocal phase-field model for non-isothermal phase transitions
with a non-conserved order parameter is studied. The paper complements
recent investigations by S. Zheng and the second author and treats
the case when the part of the free energy density forcing the order parameter
to attain values within the physically meaningful range [0; 1] is not given
by a logarithmic expression but by the indicator function of [0; 1] . The resulting
field equations form a system of integro-partial differential inclusions
that are highly nonlinearly coupled. For this system, results concerning global
existence, uniqueness and large-time asymptotic behaviour are derived. The
main results are proved by first transforming the system of inclusions into an
equivalent system of equations in which hysteresis operators occur, and then
employing techniques similar to those recently developed by the authors for
phase-field systems involving hysteresis operators.
Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility functionals
and convex risk measures. Explicit solutions are found for quantile-based coherent
risk measures and related utility functionals. Typically, these solutions exhibit a critical
phenomenon: If the capital constraint is below some critical value, then the solution will
coincide with a classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For general risk measures
and utility functionals, it is shown that there exists a solution that can be written
as a deterministic increasing function of the price density.
We study a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.
We suggest a new model for the design of telecommunication networks which integrates
decisions about the topology, configuration of the switching hardware, link dimensioning,
and protected routing of communication demands. Applying the branch-andcut-
algorithm implemented in our network planning and optimization tool discnet, we
demonstrate that real-world based network planning instances of such an enhanced model
can be solved.
In this article, we present a mathematical model and an algorithm to support one of the central
strategic planning decisions of network operators: How to organize a large number of locations into
an hierarchy of network levels? We propose a mixed-integer program and a Lagrangian relaxation
based algorithm to model and solve this planning task. As one big advantage of this approach, not
only solutions but also worst-case quality gurarantees can be provided. We present a solution for
a G-WiN planning instance of DFN with 759 locations which has been computed in less than 30
minutes and which is (provably) less than 0.5 percent away from optimality.
Der scharfeWettbewerb innerhalb der Telekommunikationsbranche zwingt die Netzbetreiber dazu,
ihre Investitionen genau zu planen und immer wieder Einsparungsmaßnahmen durchzuführen.
Gleichzeitig ist es jedoch wichtig, die Qualität der angebotenen Dienste zu verbessern, um neue
Kunden zu gewinnen und langfristig an sich zu binden.
Die mathematische Optimierung bietet sich für viele solcher Aufgabenstellungen als hervorragend
geeignetes Planungswerkzeug an. Ziel dieses Artikels ist es, ihre Methodik und ihre Anwendung
speziell zur Kosten- und Qualitätsoptimierung in Kommunikationsnetzen vorzustellen. Anhand
von vier konkreten Planungsaufgaben aus dem Bereich der Festnetzplanung wird aufgezeigt, wie
sich komplexe Zusammenhänge in flexiblen mathematischen Modellen abbilden lassen und welche
Verfahren zur automatisierten Bearbeitung der Probleme eingesetzt werden können. Die hier vorgestellten
Methoden zeichnen sich insbesondere dadurch aus, dass sie neben hochwertigen Lösungen
auch eine Qualitätsgarantie liefern, mit der sich die Lösungen fundiert bewerten lassen. Die dokumentierten
Ergebnisse aus verschiedenen Industrieprojekten belegen die Eignung und Güte der
mathematischen Optimierung für die Praxis.
This paper demonstrates simulation tools for edge-emitting multi quantum well (MQW) lasers.
Properties of the strained MQW active region are simulated by eight-band kp calculations. Then, a 2D
simulation along the transverse cross section of the device is performed based on a drift-diffusion model,
which is self-consistently coupled to heat transport and equations for the optical field. Furthermore, a
method is described, which allows for an efficient quasi 3D simulation of dynamic properties of multisection
edge-emitting lasers.
Let H be a semi–bounded self–adjoint operator in a separable Hilbert space.
For a certain class of positive, continuous, decreasing, and convex functions
F we show the convexity of trace functionals tr(F (H + U − ε(U ))) − ε(U ),
where U is a bounded self–adjoint operator on H and ε(U ) is a normalizing
real function—the Fermi level—which may be identical zero. If additionally
F is continuously differentiable, then the corresponding trace functional is
Fréchet differentiable and there is an expression of its gradient in terms off
the derivative of F . The proof of the differentiability of the trace functional
is based upon Birman and Solomyak’s theory of double Stieltjes operator
integrals. If, in particular, H is a Schrödinger–type operator and U a real-valued function, then the gradient of the trace functional is the quantum
mechanical expression of the particle density with respect to an equilibrium
distribution function f = −F . Thus, the monotonicity of the particle density
in its dependence on the potential U of Schrödinger’s operator—which has
been understood since the late 1980s—follows as a special case.
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
For a refined network analysis, we are interested in circuit simulation
including distributed models of semiconductors. We construct a mathematical
model for nonlinear electric networks containing semiconductors
described by the drift-diffusion equations. The focus lies on the coupling
of the network DAEs and the semiconductor PDEs.
Furthermore, we study the behavior of the coupled systems with respect to
time dependent perturbations using an index concept for abstract DAEs.
We present a network topological criterion that guarantees index-1 systems.
We consider the problem of utility maximization for small traders on incomplete
financial markets. As opposed to most of the papers dealing with this
subject, the investors’ trading strategies we allow underly constraints described
by closed, but not necessarily convex, sets. The final wealths obtained by trading
under these constraints are identified as stochastic processes which usually are
supermartingales, and even martingales for particular strategies. These strategies
are seen to be optimal, and the corresponding value functions determined
simply by the initial values of the supermartingales. We separately treat the
cases of exponential, power and logarithmic utility.
We consider financial markets with agents exposed to an external source of
risk which cannot be hedged through investments on the capital market alone.
The sources of risk we think of may be weather and climate. Therefore we face
a typical example of an incomplete financial market. We design a model of a
market on which the external risk becomes tradable. In a first step we complete
the market by introducing an extra security which valuates the external risk
through a process parameter describing its market price. If this parameter is
fixed, risk has a price and every agent can maximize the expected exponential
utility with individual risk aversion obtained from his risk exposure on the one
hand and his investment into the financial market consisting of an exogenous set
of stocks and the insurance asset on the other hand. In the second step, the
market price of risk parameter has to be determined by a partial equilibrium
condition which just expresses the fact that in equilibrium the market is cleared
of the second security. This choice of market price of risk is performed in the
framework of nonlinear backwards stochastic differential equations.
Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework
(2004)
We consider financial markets with agents exposed to external sources of risk
caused for example by short term climate events such as the South Pacific sea
surface temperature anomalies widely known under the name El Nino. Since
such risks cannot be hedged through investments on the capital market alone,
we face a typical example of an incomplete financial market. In order to make
this risk tradable, we use a financial market model in which an additional insurance
asset provides another possibility of investment besides the usual capital
market. Given one of many possible market prices of risk each agent can maximize
his individual exponential utility from his income obtained from trading in
the capital market, the additional security, and his risk exposure function. Under
the equilibrium market clearing condition for the insurance security the market
price of risk is uniquely determined by a backward stochastic differential equation.
We translate these stochastic equations via the Feynman-Kac formalism
into semi-linear parabolic partial differential equations. Numerical schemes are
available by which these semilinear pde can be simulated. We choose two simple
qualitatively interesting models to describe sea surface temperature, and with
an ENSO risk exposed fisher and farmer and a climate risk neutral bank three
model agents with simple risk exposure functions. By simulating the expected
appreciation price of risk trading, the optimal utility of the agents as a function
of temperature, and their optimal investment into the risk trading security we
obtain first insight into the dynamics of such a market in simple situations.
We consider financial markets with two kinds of small traders: regular traders
who perceive the asset price process S through its natural filtration, and insid-
ers who possess some information advantage which makes the filtrations through
which they perceive the evolution of the market richer. The basic question we dis-
cuss is the link between (NFLVR), the semimartingale property of S viewed from
the agent’s perspective, and bounded expected utility. We show that whenever
an agent’s expected utility is finite, S is a semimartingale with a Doob-Meyer
decomposition featuring a martingale part and an information drift. The ex-
pected utility gain of an insider with respect to a regular trader is calculated in
a completely general setting. In particular, for the logarithmic utility function,
utility gain is a function of the relative information drift alone, regardless of the
completeness of the market.
Invariant measures of dynamical systems generated e. g. by difference equations
can be computed by discretizing the originally continuum state space, and
replacing the action of the generator by the transition mechanism of a Markov
chain. In fact they are approximated by stationary vectors of these Markov
chains. Here we extend this well known approximation result and the underlying
algorithm to the setting of random dynamical systems, i.e. dynamical systems
on the skew product of a probability space carrying the underlying stationary
stochasticity and the state space, a particular non-autonomous framework. The
systems are generated by difference equations driven by stationary random processes
modelled on a metric dynamical system. The approximation algorithm
involves spatial discretizations and the definition of appropriate random Markov
chains with stationary vectors converging to the random invariant measure of
the system.
The work presents a novel method for the detection of mutual
phase synchronization in non-stationary time series. We show how the
application of a cluster algorithmthat considers spatio-temporal struc-
tures of data follows from the general condition of phase-synchronized
data. In view of the topology of phasic data, we re-formulate the
K-Means cluster algorithm on a flat torus and apply a segmentation
index derived in an earlier work (Physica D 177,203-232(2003)). This index is extended by means of averaging in order to reflect phase syn-
chronization in ensembles of multivariate time series. The method is
illustrated using simulated multivariate phase dynamics and arrays of
chaotic systems, in which temporal segments of phase-synchronized
states are registered. A comparison with results from an existing bi-
variate synchronization index reveals major advantages of our method.
A stability analysis is presented for neural field equations in the presence
of axonal delays and for a general class of connectivity kernels and synap-
tic properties. Sufficient conditions are given for the stability of equilibrium
solutions. It is shown that the delays play a crucial role in non-stationary
bifurcations of equilibria, whereas the stationary bifurcations depend only on
the kernel. Bounds are determined for the frequencies of bifurcating periodic
solutions. A perturbative scheme is used to calculate the types of bifurca-
tions leading to spatial patterns, oscillatory solutions, and traveling waves.
For high transmission speeds a simple method is derived that allows the de-
termination of the bifurcation type by visual inspection of the Fourier trans-
forms of the connectivity kernel and its first moment. Results are numerically
illustrated on a class of neurologically plausible second order systems with
combinations of Gaussian excitatory and inhibitory connections.
This work studies dynamical properties of spatially extended neu-
ronal ensembles. We first derive an evolution equation from tem-
poral properties and statistical distributions of synapses and somata.
The obtained integro-differential equation considers both synaptic and
axonal propagation delay, while spatial synaptic connectivities ex-
hibit gamma-distributed distributions. This familiy of connectivity
kernels also covers the cases of divergent, finite, and negligible self-
connections. The work derives conditions for both stationary and
nonstationary instabilities for gamma-distributed kernels.It turns out
that the stability conditions can be formulated in terms of the mean spatial interaction ranges and the mean spatial interaction times. In
addition, a numerical study examines the evoked spatiotemporal re-
sponse activity caused by short local stimuli and reveals maximum
response activity after the mean interaction time at a distance from
stimulus offset location equal to the mean interaction range. These
findings propose new insights to neuronal mechanisms of experimen-
tally observed evoked brain activity.