Nonstationary Increments Research Articles

Analysing the variability of non-stationary extreme rainfall events amidst climate change in East Malaysia

ABSTRACT Climate change is intensifying the occurrence of extreme rainfall events, drawing attention to the importance of understanding the return period concept within the realm of extreme weather studies. This study evaluates the stationarity of extreme rainfall series on both monthly and annual series across East Malaysia, employing the Augmented Dickey–Fuller, Phillips Perron, and Kwiatkowski–Phillips–Schmidt–Shin tests. To model these extreme rainfall series, various probability distributions were applied, followed by goodness-of-fit tests to determine their adequacy. The study identified the stationary and non-stationary return values at 25-, 50-, and 100-year return periods. Additionally, maps depicting the spatial distribution for non-stationary increment were generated. The results indicated that extreme monthly rainfall exhibited stationary characteristics, while extreme yearly rainfall displayed non-stationary characteristics. Among the tested probability distributions, the generalised extreme value distribution was found to be superior in representing the characteristics of the extreme rainfall. Furthermore, a significant finding is that the non-stationary rainfall exhibits higher return values than those of stationary rainfall across all return periods. The northeast coast of Sabah highlighted as the most affected area, with notably high return values for extreme rainfall.

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional parameter H ( ⋅ ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally more regular than { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} itself. On one hand this result implies that the two non-classical multifractional processes { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} have exactly the same local path behavior as that of the classical MBM { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} . On the other hand it allows to obtain from discrete realizations of { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} strongly consistent statistical estimators for values of their functional parameter.

Gaussian and hermite Ornstein–Uhlenbeck processes

In the present paper we study the asymptotic behavior of the auto-covariance function for Ornstein–Uhlenbeck (OU) processes driven by Gaussian noises with stationary and non-stationary increments and for Hermite OU processes. Our results are generalizations of the corresponding results of Cheridito et al. and Kaarakka and Salminen.

Pricing energy quanto options in the framework of Markov-modulated additive processes

Abstract Energy quanto options are risk management tools that have a payoff similar to the product of the payoffs of two options, each written on an energy-related underlying. These options, as opposed to standardized contracts that only account for price risk, are designed to manage both volumetric and price risk in energy markets. Since the use of such options enables actors in the energy market also to hedge against production volume risk, they are becoming very popular. This paper considers the valuation of such an option on futures when the underlying futures prices are governed by Markov-modulated additive processes, which have independent but non-stationary increments within each regime. We derive a valuation formula by using the Fast Fourier Transform (FFT) technique under the assumption that the joint characteristic function of the log-futures prices is known analytically. We study this approximation under different regime-switching models. Several numerical case studies illustrate that our FFT-based valuation has a high precision and is much faster than Monte Carlo estimates.

Taylor Based Jump Diffusion Model of Fractional Brownian Motion of Stock Price

Assuming that the stock price follows the stochastic differential equation driven by fractional Brownian motion and jump process, under the condition that the interest rate and volatility are constant. It has nothing to do with the relative importance of the information causing the jump. Generally speaking, the relative jump height of stock price depends on the relative importance of the important information. Fractional Brownian motion not only has various properties of fractional Brownian motion, but also has nonstationary increment. On the basis of fractional Brownian motion model, the fractional Brownian motion model with jump, namely fractional jump-diffusion process, is investigated. It is proposed to use Taylor expansion to deal with the control items in the model. For large-scale numerical simulation, it can reduce the amount of calculation required by the algorithm and save computer memory. The method of guessing value function and Taylor series expansion is introduced to solve the value function. The dialectical relationship between mathematics and finance is shown from one side: on the one hand, mathematics is a powerful tool for financial research; on the other hand, financial practice has promoted the development of mathematical theory itself.

Remaining Useful Life Prediction for Degradation Processes With Dependent and Nonstationary Increments

Remaining useful life (RUL) prediction is critical for health management of industrial equipment. It has been widely noted that degradation modeling is a core step for RUL prediction where the Brownian motion (BM)-based models attract much attention. However, the existing BM-based degradation models still have some impractical assumptions, where the increments of a BM are independent and stationary. To extend the application of the degradation models, a bifractional Brownian motion (biFBM)-based degradation model is developed in this article. The biFBM is a process with dependent and nonstationary increments, which includes the BM and fractional Brownian motion (FBM) as special cases. For the proposed degradation model, the estimation of parameters and degradation states as well as the prediction of RUL is further considered. To address the non-Markovian degradation processes, an improved particle filter is designed for degradation state estimation and RUL prediction. The proposed degradation model and RUL prediction method are validated by case studies of turbine engines and a blast furnace wall.

On stationarity properties of generalized Hermite-type processes

The paper investigates properties of generalized Hermite-type processes that arise in non-central limit theorems for integral functionals of long-range dependent random fields. The case of increasing multidimensional domain asymptotics is studied. Three approaches to investigate the properties of these processes are discussed. Contrary to the classical one-dimensional case, it is shown that for any choice of a multidimensional observation window the generalized Hermite-type process has non-stationary increments.

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

AbstractWe study shot noise processes with Poisson arrivals and nonstationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: (a) the conditional variance functions of the noises have a power law and (b) the conditional noise distributions are piecewise. In both cases, the limit processes are self‐similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self‐similar Gaussian processes with nonstationary increments, via the time‐domain integral representation, which are natural generalizations of fractional Brownian motions.

Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator

The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by . Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by using the right-continuous inverse of H, we define an inhomogeneous analog of the time-fractional Poisson process.

Limit theorems for functionals of two independent Gaussian processes

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed.

Maximum likelihood estimation for Gaussian process with nonlinear drift

We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained generalize the paper [Yu. Mishura, K. Ralchenko, S. Shklyar, Maximum likelihood drift estimation for Gaussian process with stationary increments, Austrian J. Stat., 46(3–4): 67–78, 2017] in two directions: the drift may be nonlinear, and the noise may have nonstationary increments. As an example, the model with subfractional Brownian motion is considered.

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.

Variable diffusion in stock market fluctuations

We analyze intraday fluctuations in several stock indices to investigate the underlying stochastic processes using techniques appropriate for processes with nonstationary increments. The five most actively traded stocks each contains two time intervals during the day where the variance of increments can be fit by power law scaling in time. The fluctuations in return within these intervals follow asymptotic bi-exponential distributions. The autocorrelation function for increments vanishes rapidly, but decays slowly for absolute and squared increments. Based on these results, we propose an intraday stochastic model with linear variable diffusion coefficient as a lowest order approximation to the real dynamics of financial markets, and to test the effects of time averaging techniques typically used for financial time series analysis. We find that our model replicates major stylized facts associated with empirical financial time series. We also find that ensemble averaging techniques can be used to identify the underlying dynamics correctly, whereas time averages fail in this task. Our work indicates that ensemble average approaches will yield new insight into the study of financial markets’ dynamics. Our proposed model also provides new insight into the modeling of financial markets dynamics in microscopic time scales.

Large Deviation Multifractal Analysis of a Class of Additive Processes With Correlated Nonstationary Increments

We consider a family of stochastic processes built from infinite sums of independent positive random functions on \BBR+. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on \BBR+. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic. Such processes bear some analogy with Levy processes, but are more complex since their increments are neither stationary nor independent. In the work of Barral and Levy Vehel, the Hausdorff multifractal spectrum of these processes was computed. We are interested here in their Large Deviation and Legendre multifractal spectra. These “statistical” spectra are seen to give, in this case, a richer information than the “geometrical” Hausdorff spectrum. In addition, our results provide a firm theoretical basis for the empirical discovery of the multifractal nature of TCP traffic.

Ubiquity of, and geostatistics for, nonstationary increment random fields

[1] Nonstationary random fields such as fractional Brownian motion and fractional Levy motion have been studied extensively in the hydrology literature. On the other hand, random fields that have nonstationary increments have seen little study. A mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant. The abundance of nonstationary increment processes has important implications, e.g., in kriging where a translation-invariant variogram implicitly assumes stationarity of the increments. An approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results.

Two-scale renormalization-group classification of diffusive processes

Renormalization-group operators are used to classify stochastic processes on two time scales. Repeated application of one operator is associated with the long-time behavior of the process while the other is associated with the short-time behavior of the process. This approach is shown to be robust even in the presence of nonstationary increments and infinite second moments. Fixed points of the operators can be used for further subclassification of processes when appropriate limits exist. Several processes are classified using the renormalization-group scheme. The processes to be classified include advection-diffusion in an ergodic velocity field, and a model of diffusion in the human bronchial tree.

Ensemble vs. time averages in financial time series analysis

Empirical analysis of financial time series suggests that the underlying stochastic dynamics are not only non-stationary, but also exhibit non-stationary increments. However, financial time series are commonly analyzed using the sliding interval technique that assumes stationary increments. We propose an alternative approach that is based on an ensemble over trading days. To determine the effects of time averaging techniques on analysis outcomes, we create an intraday activity model that exhibits periodic variable diffusion dynamics and we assess the model data using both ensemble and time averaging techniques. We find that ensemble averaging techniques detect the underlying dynamics correctly, whereas sliding intervals approaches fail. As many traded assets exhibit characteristic intraday volatility patterns, our work implies that ensemble averages approaches will yield new insight into the study of financial markets’ dynamics.

On generating conductivity fields with known fractal dimension and nonstationary increments

Fractional Brownian motion (fBm) is a stochastic process that has stationary increments with long‐range correlations and known fractal dimension. We study a multiple‐dimensional extension of fBm with nonstationary increments that allows for trends in the statistical structure while maintaining the Gaussian nature and fractal dimension of fBm. Two methods for simulating this extension are employed and described in detail. One approach combines Cholesky decomposition with a generalization of random midpoint displacement. The other makes repeated use of the Cholesky decomposition. The resulting fields can be employed in various geophysical settings, e.g., as log conductivity fields in hydrology and topographic elevation in geomorphology.

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like tβ with β 1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Levy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.