Hilbert Spaces Formed by Strongly Harmonizable Stable Processes

2001 ◽  
Vol 8 (1) ◽  
pp. 181-188
Author(s):  
A. R. Soltani ◽  
B. Tarami
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Continuous Time ◽  
Moving Average ◽  
Stable Process ◽  
Stable Processes ◽  
Moving Average Representation ◽  
Process Elements ◽  
Average Representation

Abstract A strongly harmonizable continuous time symmetric α-stable process is considered. By using covariations, a Hilbert space is formed from the process elements and used for a purpose of moving average representation and prediction.

Noncausality in Continuous Time Models

1996 ◽  
Vol 12 (2) ◽  
pp. 215-256 ◽  
Author(s):  
F. Comte ◽  
E. Renault
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Stochastic Volatility Models ◽  
Discrete Sample ◽  
Volatility Models ◽  
Moving Average Representation ◽  
Canonical Representations ◽  
Average Representation

In this paper, we study new definitions of noncausality, set in a continuous time framework, illustrated by the intuitive example of stochastic volatility models. Then, we define CIMA processes (i.e., processes admitting a continuous time invertible moving average representation), for which canonical representations and sufficient conditions of invertibility are given. We can provide for those CIMA processes parametric characterizations of noncausality relations as well as properties of interest for structural interpretations. In particular, we examine the example of processes solutions of stochastic differential equations, for which we study the links between continuous and discrete time definitions, find conditions to solve the possible problem of aliasing, and set the question of testing continuous time noncausality on a discrete sample of observations. Finally, we illustrate a possible generalization of definitions and characterizations that can be applied to continuous time fractional ARMA processes.

scholarly journals Ruin probability with certain stationary stable claims generated by conservative flows

2007 ◽  
Vol 39 (02) ◽  
pp. 360-384 ◽  
Author(s):  
Uğur Tuncay Alparslan ◽  
Gennady Samorodnitsky
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Ruin Probability ◽  
Stable Process ◽  
Independent Interest ◽  
Stable Processes ◽  
Auxiliary Result ◽  
Initial Capital ◽  
Order Of Magnitude

We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.

scholarly journals Exceedance of power barriers for integrated continuous-time stationary ergodic stable processes

2009 ◽  
Vol 41 (03) ◽  
pp. 874-892
Author(s):  
Uğur Tuncay Alparslan
Keyword(s):  
Continuous Time ◽  
Stable Process ◽  
Tail Probability ◽  
Exceedance Probability ◽  
Dependence Structure ◽  
Stable Processes ◽  
Limiting Behavior ◽  
Order Of Magnitude ◽  
The Difference ◽  
Conservative Flow

We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable processes generated by ergodic dissipative flows is established. In the second part an example with the integral of a stable process generated by a conservative flow is analyzed. Finally, the difference in the order of magnitude of the exceedance probability in the two cases is related to the dependence structure of the underlying stable process.

Solutions of Linear Rational Expectations Models

1985 ◽  
Vol 1 (3) ◽  
pp. 341-368 ◽  
Author(s):  
L. Broze ◽  
C. Gourieroux ◽  
A. Szafarz
Keyword(s):  
Rational Expectations ◽  
Moving Average ◽  
Stationary Solutions ◽  
Global Approach ◽  
Number Of Solutions ◽  
Simultaneous Treatment ◽  
Moving Average Representation ◽  
Natural Parametrization ◽  
Average Representation ◽  
Independent Parameters

Linear rational expectations models generally have a large number of solutions. It is thus important to describe them exhaustively in order to study their properties and subsequently estimate which solution best fits the data. In this paper, a global approach is suggested allowing a simultaneous treatment of all possible cases. The fundamental concepts are the revision processes appearing in the procedure of updating expectations. It isfound that the set of solutions is completely described by using a limitednumber of these processes. We show how the method may be applied to determine the set of stationary solutions admitting an infinite moving-average representation. We give a natural parametrization of this set and discuss the exact number of independent parameters.

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

1984 ◽  
Vol 16 (4) ◽  
pp. 819-842 ◽  
Author(s):  
K. F. Turkman ◽  
A. M. Walker
Keyword(s):  
Moving Average ◽  
Random Coefficients ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Trigonometric Polynomials ◽  
Stationary Gaussian Sequence ◽  
Moving Average Representation ◽  
Standard Normal ◽  
Average Representation ◽  
Mutually Independent

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

A REPRESENTATION THEORY FOR POLYNOMIAL COFRACTIONALITY IN VECTOR AUTOREGRESSIVE MODELS

2009 ◽  
Vol 26 (4) ◽  
pp. 1201-1217 ◽  
Author(s):  
Massimo Franchi
Keyword(s):  
Representation Theory ◽  
Autoregressive Model ◽  
Moving Average ◽  
Recursive Algorithm ◽  
Autoregressive Models ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive ◽  
Moving Average Representation ◽  
Average Representation

We extend the representation theory of the autoregressive model in the fractional lag operator of Johansen (2008, Econometric Theory 24, 651–676). A recursive algorithm for the characterization of cofractional relations and the corresponding adjustment coefficients is given, and it is shown under which condition the solution of the model is fractional of order d and displays cofractional relations of order d − b and polynomial cofractional relations of order d − 2b,…, d − cb ≥ 0 for integer c; the cofractional relations and the corresponding moving average representation are characterized in terms of the autoregressive coefficients by the same algorithm. For c = 1 and c = 2 we find the results of Johansen (2008).

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