Alexander Lindner

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

Continuous-time GARCH processes

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data.

Continuous Time Volatility Modelling: COGARCH versus Ornstein–Uhlenbeck Models

We compare the probabilistic properties of the non-Gaussian Ornstein-Uhlenbeck based stochastic volatility model of Barndorff-Nielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be shared by both processes, but differences are pointed out as well. Furthermore, it is shown that the COGARCH process has Pareto like tails under weak regularity conditions.

Method of Moment Estimation in the Cogarch(1,1) Model

We suggest moment estimators for the parameters of a continuous time GARCH(1,1) process based on equally spaced observations. Using the fact that the increments of the COGARCH(1,1) process are strongly mixing with exponential rate, we show that the resulting estimators are consistent and asymptotically normal. We investigate the empirical quality of our estimators in a simulation study based on the variance gamma driven COGARCH(1,1) model. The estimated volatility with corresponding residual analysis is also presented. Finally, we fit the model to high-frequency data. Copyright Royal Economic Society 2007

On continuity properties of the law of integrals of levy processes

Let (ξ, η) be a bivariate Levy process such that the integral \(\int_0^\infty {e^{ - \xi _{t - } } d\eta _t }\)converges almost surely. We characterise, in terms of their Levy measures, those Levy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I := ∫ 0 ∞ g(ξ t ) dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ 0 ∞ g(ξ t ) dY t , where Y is an almost surely strictly increasing stochastic process, independent of ξ.

Extremal behavior of stochastic volatility models

Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has — sometimes quite substantial — upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Levy driven volatility processes as, for instance, by Levy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Levy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Levy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels.

Continuous Time Approximations to GARCH and Stochastic Volatility Models

We collect some continuous time GARCH models and report on how they approximate discrete time GARCH processes. Similarly, certain continuous time volatility models are viewed as approximations to discrete time volatility models.

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Properties of the law μ of the integral ∫ ∞ 0 c -N t- dY t are studied, where c > 1 and {(N t , Y t ), t ≥ 0) is a bivariate Levy process such that {N t } and {Y t } are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein-Uhlenbeck process. The law μ is parametrized by c, q and r, where p = 1 - q - r, q, and r are the normalized Levy measure of {(N t , Y t )} at the points (1,0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p > 0 and q > 0, μ c,q,r is infinitely divisible if and only if r ≤ pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r = 1. It is shown that if c is in the set of Pisot-Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q > 0. On the other hand, for Lebesgue almost every c > 1, there are positive constants C 1 and C 2 such that 11 is absolutely continuous whenever q ≥ C 1 p ≥ C 2 r. For any c > 1 there is a positive constant C 3 such that μ is continuous-singular whenever q > 0 and max{q, r} ≤ C 3p . Here, if {N t } and {Y t } are independent, then r = 0 and q = b/(a + b).

Frames of exponentials: lower frame bounds for finite subfamilies and approximation of the inverse frame operator

Abstract We give lower frame bounds for finite subfamilies of a frame of exponentials { e i λ k (·) } k∈ Z in L2(−π,π). We also present a method for approximation of the inverse frame operator corresponding to { e i λ k (·) } k∈ Z , where knowledge of the frame bounds for finite subfamilies is crucial.

Asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors

We derive results on the asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors. They appear in particular in delta-gamma models in financial risk management approximating portfolio returns. Quantile estimation corresponds to the estimation of the Value-at-Risk, which is a serious problem in high dimension.