1,720,981 research outputs found
Extremes of subexponential Lévy driven moving average processes. Preprint, available at http://www.ma.tum.de/stat
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Statistical estimation of multivariate Ornstein-Uhlenbeck processes and applications to co-integration
Full text linkAbstract Ornstein-Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein-Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time. JEL Classifications: C13, C2
Extremes of Regularly Varying Lévy Driven Mixed Moving Average Processes
No full textIn this paper we study the extremal behavior of stationary mixed moving average processes Y (t) = � R+×R f(r, t − s) dΛ(r, s) for t ∈ R, where f is a deterministic function and Λ is an infinitely divisible independently scattered random measure, whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of certain functionals of Y. The extremal behavior is modelled by marked point processes at a discrete-time skeleton chosen properly by the jump times of the underlying driving Lévy process and the extremes of the kernel function. The sequences of marked point processes converge weakly to a cluster Poisson random measure and reflect extremes of Y on a high level. We obtain also convergence of partial maxima to the Fréchet distribution. Our models and results cover short and long range dependence regimes
Extremes of Lévy Driven Mixed MA Processes with Convolution Equivalent Distributions
No full textWe investigate the extremal behavior of stationary mixed MA processes 0, where f is a deterministic function with f(r, 0) = f s)| for s 0 and r R+ . The random measure # is infinitely divisible and independently scattered, whose finite dimensional distributions, represented by L(1) = #(R+ [0, 1]), are in the class of convolution equivalent distributions and in the maximum domain of attraction of the Gumbel distribution. It is shown that the tail of the stationary distribution of Y decreases faster to 0 than the tail of f L(1). In contrast to this the tail of the maximum of Y over a fixed time interval decreases of the same order of magnitude as the tail of f L(1) and is linearly in the length of the interval. We divide the positive real line into properly chosen randomly intervals and denote the maxima of the process in these intervals by (M k ) k#N . The extremal behavior of Y is completely described by a weak limit of marked point processes based on (M k ) k#N . A complementary result guarantees the convergence of running maxima of Y to the Gumbel distribution. AMS 2000 Subject Classifications: primary: 60G70 secondary: 60F05, 60G10, 60G55 Keywords: convolution equivalent distribution, extreme value theory, MA process, marked point process, mixed MA process, point process, random measure, shot noise process, subexponential distribution, supOU process # Center for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany, email: [email protected], www.ma.tum.de/stat/
Dependence Estimation for High Frequency Sampled Multivariate CARMA Models
No full textThe paper considers high frequency sampled multivariate continuous-time ARMA (MCARMA) models, and derives the asymptotic behavior of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behavior of the crosscovariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous-time and in the discrete-time model. As special case we consider a CARMA (one-dimensional MCARMA) process. For a CARMA process we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models, only the sums in the discretetime model are exchanged by integrals in the continuous-time model. Finally, we present limit results for multivariate MA processes as well which are not known in this generality in the multivariate setting yet
Extremes of subexponential Lévy driven moving average processes. Preprint, available at http://www.ma.tum.de/stat
No full textIn this paper we study the extremal behavior of a stationary continuoustime moving average process Y (t) = � ∞ − ∞ f(t − s) dL(s) for t ∈ R, where f is a deterministic function and L is a Lévy process whose increments, represented by L(1), are subexponential and in the maximum domain of attraction of the Gumbel distribution. We give necessary and sufficient conditions for Y to be a stationary, infinitely divisible process, whose stationary distribution is subexponential, and in this case we calculate its tail behavior. We show that large jumps of the Lévy process in combination with extremes of f cause excesses of Y and thus properly chosen discrete-time points are sufficient to specify the extremal behavior of the continuous-time process Y. We describe the extremal behavior of Y completely by a weak limit of marked point processes. A complementary result guarantees the convergence of running maxima of Y to the Gumbel distribution
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