34 research outputs found
On Observability of Chaotic Systems: An Example
The concept of observability of a special chaotic system, namely the dyadic map, is studied here in case the observation is not exact. The usual concept of observable subspace does not distinguish among the behavior of different models. It turns out that a suitable extension of this concept can be obtained using the idea of Hausdorff dimension. It is shown that this dimension increases as the observation error becomes smaller, and is equal to one only if the system is observable
Derivative pricing for a multi-curve extension of the Gaussian, exponentially quadratic short rate model
The recent financial crisis has led to so-called multi-curve models for the
term structure. Here we study a multi-curve extension of short rate models
where, in addition to the short rate itself, we introduce short rate spreads.
In particular, we consider a Gaussian factor model where the short rate and the
spreads are second order polynomials of Gaussian factor processes. This leads
to an exponentially quadratic model class that is less well known than the
exponentially affine class. In the latter class the factors enter linearly and
for positivity one considers square root factor processes. While the square
root factors in the affine class have more involved distributions, in the
quadratic class the factors remain Gaussian and this leads to various
advantages, in particular for derivative pricing. After some preliminaries on
martingale modeling in the multi-curve setup, we concentrate on pricing of
linear and optional derivatives. For linear derivatives, we exhibit an
adjustment factor that allows one to pass from pre-crisis single curve values
to the corresponding post-crisis multi-curve values
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
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An explicit state-space approach to the one-block super-optimal distance problem
An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115–1134, 1993) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960–982, 1993; Limebeer et al. in Int J Control 50(6):2431–2466, 1989). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control