scholarly journals Aliasing Free For Mixed Spectra for Stable Processes

2021 ◽  
Author(s):  
Rachid Sabre
Keyword(s):  
Spectral Density ◽  
Continuous Time ◽  
Spectral Measure ◽  
Polynomial Kernel ◽  
Stable Processes ◽  
Discrete Observations ◽  
Continuous Part ◽  
Discrete Measure ◽  
Alpha Stable Processes

This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.

scholarly journals Mixed Spectra for Stable Signals from Discrete Observations

2021 ◽  
Vol 12 (05) ◽  
pp. 21-44
Author(s):  
Rachid Sabre
Keyword(s):  
Spectral Density ◽  
Continuous Time ◽  
Continuous Measurement ◽  
Spectral Measurement ◽  
Polynomial Kernel ◽  
Time Process ◽  
Discrete Observations ◽  
Continuous Part ◽  
Continuous Time Process ◽  
Discrete Measurement

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

scholarly journals Some Applications of Lévy Processes to Stochastic Investment Models for Actuarial Use

1998 ◽  
Vol 28 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Investment Model ◽  
Stable Processes ◽  
Gamma Processes ◽  
Investment Models

AbstractThis paper presents a continuous time version of a stochastic investment model originally due to Wilkie. The model is constructed via stochastic differential equations. Explicit distributions are obtained in the case where the SDEs are driven by Brownian motion, which is the continuous time analogue of the time series with white noise residuals considered by Wilkie. In addition, the cases where the driving “noise” are stable processes and Gamma processes are considered.

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 Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

2012 ◽  
Vol 49 (3) ◽  
pp. 806-820
Author(s):  
Pieter C. Allaart
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Analogous Result ◽  
Stopping Time ◽  
Stable Processes ◽  
Finite Variation ◽  
Lévy Measure ◽  
One Dimensional ◽  
Stationary Increments ◽  
Reverse Relationship

Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

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.

Sign in / Sign up

Export Citation Format

Share Document