Alpha-stable Processes Research Articles

Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds

Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided is anticipative enough, despite the process featuring infinite marginal variance. Formulas of these moments at any forecast horizon under any admissible parameterization are provided. Under the assumption of errors with regularly varying tails, closed-form formulas of the predictive distribution during explosive bubble episodes are obtained and expressions of the ex ante crash odds at any horizon are available. It is found that the noncausal autoregression of order 1 (AR(1)) with AR coefficient ρ and tail exponent α generates bubbles whose survival distributions are geometric with parameter . This property extends to bubbles with arbitrarily shaped collapse after the peak, provided the inflation phase is noncausal AR(1)-like. It appears that mixed causal–noncausal processes generate explosive episodes with dynamics à la Blanchard and Watson which could reconcile rational bubbles with tail exponents greater than 1.

Controlling the Rotor Angle Stability of Single Machine Infinite Bus System in the Presence of Wiener and Alpha-Stable Levy Type Power Fluctuations

The integration of renewable energy sources into the power systems and the growth of electricity consumption leads to a considerable increase in the power fluctuations. In the first part of this study, the control of the rotor angle stability of single machine infinite bus system in the presence of Wiener type power fluctuations has been achieved by minimizing the corresponding stochastic sensitivity function. In the second part, the power fluctuations have been modeled by alpha-stable Levy processes and since stochastic sensitivity function is not available for alpha-stable Levy processes, then the control of the rotor angle stability has been numerically achieved by minimizing the corresponding rotor angle dispersion for the first time in the literature.

The Choice of the Smoothing Parameter for Alpha Stable Signals

In this work we consider the class of symmetric alpha stable processes which are a particular family of processes with infinite energy. These processes used in modeling the random signals with indefinitely growing variance. The spectral density estimator of such signals is given in the literature by smoothing the periodogram by a spectral window. Thus, the estimator depends on the width of the spectral window considered as a smoothing parameter. The choice of this parameter plays an important role since the rate of convergence of the estimator is a function of this parameter. The objective of this paper is to propose a method giving the optimal parameter based on the cross validation technique (minimization of MISE: Mean Integrate Square of Error). We establish a criterion function and we prove that the mean of this criterion converges to MISE. Thus, we show that the value minimizing this criterion is the optimal smoothing parameter. The rate of convergence of the estimator has been studied in order to prove that the smoothing parameter obtained by this method gives the fastest convergence of the estimator towards the spectral density.

Aliasing Free and Additive Error in Spectra for Alpha Stable Signals

This work focuses on the symmetric alpha stable process with continuous time frequently used in modeling the signal with indefinitely growing variance, often observed with an unknown additive error. The objective of this paper is to estimate this error from discrete observations of the signal. For that, we propose a method based on the smoothing of the observations via Jackson polynomial kernel and taking into account the width of the interval where the spectral density is non-zero. This technique allows avoiding the “Aliasing phenomenon” encountered when the estimation is made from the discrete observations of a process with continuous time. We have studied the convergence rate of the estimator and have shown that the convergence rate improves in the case where the spectral density is zero at the origin. Thus, we set up an estimator of the additive error that can be subtracted for approaching the original signal without error.

A novel adaptive beamforming algorithm against impulsive noise with alpha-stable process for satellite navigation signal acquisition

Adaptive beamforming is an effective spatial filtering technique for overcoming the vulnerability to interference of global navigation satellite system (GNSS) receivers. Although beamforming-based satellite system has the capability of nulling electronic interference sources, the distortions to GNSS receiver induced by impulsive noises are always neglected. This paper addresses the satellite navigation signal acquisition problem in the presence of impulsive noises with alpha-stable noise using the maximum correntropy criterion in framework of the GNSS system. In addition, in order to decrease the number of active elements for avoiding overmuch energy consumption, a sparse regularization is introduced to the constraints of the novel criterion. From the analysis, the novel constraint sparse maximum correntropy (CSMC) beamforming technique that can achieve robustness against impulsive noises which uses less power is developed in this manuscript for satellite signal acquisition. The proposed CSMC, maintains the robustness against impulsive outliers and achieve better performance in conjunction with less power consumption. A mean square analysis of the CSMC algorithm is presented to verify the validity of our theory. Simulation results demonstrate the superiority of the proposed methods over other previously developed beamforming techniques in GNSS.

On a Lower Asymptotic Bound of the Overflow Probability in a Fluid Queue with a Heterogeneous Fractional Input

For a fluid queue fed by superposition of fractional Brownian motion and alpha-stable Levy process, the asymptotic lower bound of the overflow probability is obtained.

Spectral Density Estimate for Stable Processes Observed with an Additive Error

In this paper, a symmetric alpha stable process where its spectral representation has an additive error is considered. The error is supposed to be constant. A periodogram as estimator of the spectral density and its rate of convergence are given. In order to give an asymptotically unbiased and consistent estimate of the spectral density, this periodogram is smoothed by an adapted spectral window. The rate of convergence is given.

Transition density estimates for diagonal systems of SDEs driven by cylindrical alpha-stable processes

We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and H\older continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove H\older and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.

Multilevel Monte Carlo for exponential L\xe9vy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and alpha-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.

Weak Moment of a Class of Stochastic Heat Equation with Martingale-valued Harmonic Function

A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + \sigma(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a L\'{e}vy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise exponential rate for the case of $\mathcal{L}=-(-\Delta)^{\alpha/2},\,\,\alpha\in(1,2]$ generator of an alpha-stable process.

Spectral Characterization of the Non-Independent Increment Family of Alpha-Stable Processes that Generalize Gaussian Process Models.

The characterization of spatial or temporal processes by a family of sufficient functions such as via a unique spectral representation and a mean function forms the basis of a large number of statistical modelling approaches. For instance, in Gaussian process regression modelling, the mean and covariance function specify uniquely the properties of the resulting statistical model. One can therefore parameterize such regression models and understand their structure and attributes generally via a specification of the covariance kernel. In this paper we generalize significantly the class of available spatial and temporal processes that may be used in statistical applications like regressions to allow for non-stationarity and heavy-tailedness. Importantly, we are able to demonstrate for the first time how to achieve this through a novel formulation no longer requiring independence of the increments in the stochastic construction of the process and statistical model. Furthermore, we achieve this in a manner akin to the covariance kernel specification in a Gaussian process model, we develop a novel covariation spectral representation of some non-stationary and non-indepenent increments symmetric Alpha-stable processes (SalphaS). Such a representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence and should allow a very wide class of statistical regression models to be subsequently developed. We present a general framework for sufficient conditions to characterize such processes and develop general constructive approaches to building models satisfying these conditions.

Supplement to: 'Spectral Characterization of the Family α-Stable Processes that Generalize Gaussian Process Models.'

Proofs and Technical Appendix for paper: Spectral characterization of the non-independent increment family of Alpha-Stable processes that generalize Gaussian process models.

ARFIMA Models and the Hurst Measures: An Investigation of Commodity Daily Index and Futures Prices

In this paper, we study some important stylized facts of commodity daily index and future prices. Our study shows that, except few, all price processes have unit roots with stationary, skewed and kurtotic increments whose R/S and generalized Hurst exponents are greater than 1/2. As a result, ARFIMA(p-1,1 d,q) models with skewed and kurtotic shocks can be good candidates for modelling commodity daily index and future prices. On the other hand, since non-linear models are practically very difficult to implement, we study linear models with fat tail shock processes. We have observed that, even though linear models with fat tail shock processes cannot generate long memory time series, they can generate time series with generalized Hurst exponents similar to ones from the commodity daily index and future prices. This suggests that with regards to the generalized Hurst exponent, linear ARIMA(p-1,1,q) models with alpha-stable shock processes are good models with statistical characteristics very similar to commodity daily prices.

Wireless Single Cellular Coverage Boundary Models

To satisfy the requirement of high wireless transmission rate from user terminals, the multi-cell cooperative communication is an important solution for cellular networks. The selection of cooperative cells is usually depended on every wireless single cellular coverage boundary in cellular networks. In this paper, we analyze the characteristics of wireless single cellular coverage boundary based on real measured data. Moreover, the alpha-stable processes are first proposed to model the wireless single cellular coverage boundary. Simulation results indicate that probability density functions of wireless single cellular coverage boundary can be fitted by alpha-stable processes with 95% confidence interval based on the real measured data. This result provides a basis for cooperative communications considering the anisotropic fading characteristic of wireless signal propagation in wireless single cellular coverage region.

Truncated Realized Covariance When Prices Have Infinite Variation Jumps

The speed of convergence of the truncated realized covariance to the integrated covariation between the Brownian parts of two semimartingales is heavily influenced by the presence of infinite activity jumps with infinite variation. Namely, the two small jumps processes play a crucial role through their degree of dependence, other than through their jump activity indices. This theoretical result is established when the marginal small jumps constitute alpha-stable processes and the semimartingales are observed discretely on a finite time horizon. The estimator in many (but not all) cases is less efficient than when the model only has finite variation jumps.The result of this paper is relevant in financial economics, since by the truncated realized covariance it is possible to separately estimate the common jumps among two assets, which has important implications in risk management and contagion modeling.

Laws of the iterated logarithm for self-normalised Lévy processes at zero

We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t ↓ 0 t\downarrow 0 ) for the “self-normalised” process ( X t − a t ) / V t (X_{t}-at)/\sqrt {V_{t}} , t > 0 t>0 , constructed from a Lévy process ( X t ) t ≥ 0 (X_{t})_{t\geq 0} having quadratic variation process ( V t ) t ≥ 0 (V_{t})_{t\geq 0} , and an appropriate choice of the constant a a . We apply them to obtain LILs when X t X_{t} is in the domain of attraction of the normal distribution as t ↓ 0 t\downarrow 0 , when X t X_{t} is symmetric and in the Feller class at 0, and when X t X_{t} is a strictly α − \alpha - stable process. When X t X_{t} is attracted to the normal distribution, an important ingredient in the proof is a Cramér-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.

Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for alpha-stable processes.

Fine regularity of Lévy processes and linear (multi)fractional stable motion

In this work, we investigate the fine regularity of Levy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Levy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

A novel covariation based noncircular sources direction finding method under impulsive noise environments

We extend the bearing estimation method for noncircular signals to the impulsive noise scenario which can be modeled as a complex symmetric alpha-stable (SαS) process. We define the extended covariation based matrix of the sensor outputs and show that it can be applied with subspace techniques to extract the bearing information from noncircular sources. Comprehensive simulations demonstrate that the proposed direction finding algorithm outperforms the traditional NC-MUSIC algorithm in the presence of a wide range of impulsive noise environments.

Particle Filtering for Acoustic Source Tracking in Impulsive Noise With Alpha-Stable Process

NonGaussian impulsive noises distort the source signal and cause problems for direction of arrival (DOA) estimation of an acoustic source. In this paper, a Bayesian framework and its particle filtering (PF) implementation for DOA tracking in the presence of complex symmetric alpha-stable noise process are developed. A constant velocity model is employed to model the source dynamics, and spatial spectra are exploited to formulate a pseudo likelihood of particles. Since the second-order statistics of alpha-stable processes do not exist, the fractional lower order moment matrix of the received data is used to replace the covariance matrix in calculating the spatial spectra. The noise usually spreads and distorts the mainlobe of the likelihood function and the particles cannot be weighted accurately. Hence, the likelihood function is exponentially weighted to emphasize the particles in a high likelihood area and thus enhance the resampling efficiency. The performance of the proposed tracking algorithm is extensively studied under simulated alpha-stable noise environments. The results show that the proposed algorithm significantly outperforms the existing PF tracking approach and the traditional localization approaches in DOA estimation.