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.
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.
We study perturbations of a stochastic program with a probabilistic constraint and r-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linear-quadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint.
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.
The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.
The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one dimensional parametrized stochastic differential equation with point delay, basic properties whereof we check.
With a view to capturing the effective dynamics and, in particular, the resonance-like behaviour of the reference model we construct a simplified or reduced model, the two state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn
can be used to characterize the phenomenon of noise-induced resonance.
Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two state model and the reference model. The resonance characteristic developed for the reduced model can thus be applied to the original model.
The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one dimensional parametrized stochastic differential equation with point delay, basic properties whereof we check. With a view to capturing the effective dynamics and, in particular, the resonance-like behaviour of the reference model we construct a simplified or reduced model, the two state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance. Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two state model and the reference model. The resonance characteristic developed for the reduced model can thus be applied to the original model.
This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical ‘delta hedge’ in complete markets.
We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect tot the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).
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.
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.
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 present a new iterative procedure for solving the multiple stopping
problem in discrete time and discuss the stability of the algorithm.
The algorithm produces monotonically increasing approximations of the
Snell envelope, which coincide with the Snell envelope after finitely many
steps. Contrary to backward dynamic programming, the algorithm allows
to calculate approximative solutions with only a few nestings of conditional
expectations and is, therefore, tailor-made for a plain Monte-Carlo
implementation.
In this project we propose the use of some widespread prediction techniques in the last few years for modeling derivatives. In order to do that, we have reviewed the state-of-the-art of the prediction models dealing with stochastic processes. In the oil futures sector, Schwartz suggested a model in which the oil futures price was split in two factors: the long-term equilibrium price and the short-term variations. As a result, we propose a Hull-White discrete-time two-factor interest rate model, whose factors are the short and the long term.
Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay
(2004)
The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback
mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays.
We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback
range and the feedback delay for general axonal and feedback connectivity kernels. In addition, we
find a novel shape of traveling fronts due to the applied feedback and criteria for its occurence are
derived.