The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the celebrated Central Limit Theorem tells us that (Yn) converges in distribution to a standard normal. The class of random variables which appear as limits in distribution of (Yn) for more general (an) and (b„) are called the stable random variables and they have a number of elegant properties, such as the preservation of their distribution functions under convolution. A stochastic process is said to be stable if all its finite-dimensional distributions are themselves stable and it is the systematic study of such processes which is the aim of the present volume.

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

In this study we use the fractional Levy stable motion (fLsm) to establish a finite iterative forecasting model with Long Range Dependent (LRD) characteristics. The LRD forecasting model considers the influence of current and past trends in stochastic sequences on future trends. We find that the discussed model can accurately forecast the trends of stochastic sequences. This fact enables us to introduce the fLsm as the fractional-order model of Levy stable motion. Self-similarity and LRD characteristics of the flsm model is introduced by using the relationship between self-similar index and the characteristic index. Thus, the order Stochastic Differential Equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The forecasting model with the LRD characteristics was obtained by discretization of FSDE. The Monte Carlo method was applied to demonstrate the feasibility of the forecasting model. The power load forecasting history data demonstrates the advantages of our model.

Fractional Lévy stable motion: Finite difference iterative forecasting model

Starting from the moving average representation of fractional Brownian motion fractional L evy processes have been constructed by keeping the same moving average kernel and replacing the Brownian motion by a pure jump L evy process with nite second moments. Another way was to replace the Brownian motion by an alpha-stable L evy process and the exponent in the kernel by H 1= . We now provide a unifying approach taking kernels of the form a((t s) + ( s) +) +b((t s) ( s) ), where can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying L evy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g. regularity of the sample paths and the semimartingale property.

/pdf/a-unifying-approach-to-fractional-levy-processes-3dai3y2nsm.pdf

A unifying approach to fractional Lévy processes

In this paper we introduce the well-balanced Levy driven Ornstein-Uhlenbeck process as a moving average process of the form $X_t=\int \exp(-\lambda |t-u|)dL_u$. In contrast to Levy driven Ornstein-Uhlenbeck processes the well-balanced form possesses continuous sample paths and an autocorrelation function which is decreasing not purely exponential but of the order $\lambda |u|\exp(-\lambda |u|)$. Furthermore, depending on the size of $\lambda$ it allows both for positive and negative correlation of increments. We indicate how the well-balanced Ornstein-Uhlenbeck process might be used as mean or volatility process in stochastic volatility models.

Well-balanced Levy Driven Ornstein-Uhlenbeck Processes

Multivariate Levy-driven mixed moving average (MMA) processes of the type X, = JJ f{A, t — s)A(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Levy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

Tail behavior of multivariate levy-driven mixed moving average processes and supou

Various characterizations for fractional Levy process to be of finite variation are obtained, one of which is in terms of the characteristic triplet of the driving Levy process, while others are in terms of differentiability properties of the sample paths. A zero-one law and a formula for the expected total variation is also given.

Finite Variation of Fractional Levy Processes

Various characterizations for fractional Levy processes to be of finite variation are obtained, one of which is in terms of the characteristic triplet of the driving Levy process, while others are in terms of differentiability properties of the sample paths. A zero-one law and a formula for the expected total variation are also given.

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.050/people/lindner/bender-lindner-schicks1.pdf

Finite Variation of Fractional Lévy Processes

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Finite Variation of Fractional Levy Processes

Finite Variation of Fractional Lévy Processes