Independent Increment Processes Research Articles

Regulating stochastic clocks§

Stochastic clocks represent a class of time change methods for incorporating trading activity into continuous-time financial models, with the ability to deal with typical asymmetrical and tail risks in financial returns. In this paper we propose a significant improvement of stochastic clocks for the same objective but without decreasing the number of trades or changing the trading intensity. Our methodology targets any Lévy subordinator, or more generally any process of nonnegative independent increments, and is based on various choices of regulating kernels motivated from repeated averaging. By way of a hyperparameter linked to the degree of regulation, arbitrarily large skewness and excess kurtosis of returns can be easily achieved. Generic-time Laplace transforms, characterizing triplets, and cumulants of the regulated clocks and subsequent mixed models are analyzed, serving purposes ranging from statistical estimation and option price calibration to simulation techniques. Under specified jump–diffusion processes and tempered stable processes, a robust moment-based estimation procedure with profile likelihood is developed and a comprehensive empirical study involving S&P500 and Bitcoin daily returns is conducted to demonstrate a series of desirable effects of the proposed methods.

Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes

In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results.

Uncertain Independent Increment Processes are Contour Processes

Uncertain processes are used to model dynamic indeterminate systems associated with human uncertainty, and uncertain independent increment processes are a type of uncertain processes with independent uncertain increments. This paper mainly verifies a basic property about the sample paths of uncertain independent increment processes, which states that uncertain independent increment processes defined on a continuous uncertainty space are contour processes, a type of uncertain processes with a spectrum of sample paths as the skeletons. Based on this property, the extreme values and the time integral of an uncertain independent increment process are investigated, and their inverse uncertainty distributions are obtained.

Conditionally independent increment processes for modeling electricity prices with regard to renewable power generation

Renewable energies have become responsible for a large part of the variation in electricity prices. We offer a new approach that makes it possible to represent this dependency in a flexible way. This allows to vary especially price spikes both in direction and size according to the generation of wind and solar power. The proposed concept is based on the construction of a process with conditionally independent increment. We provide an estimation procedure for them and offer a test to verify the dependency on an external variable. Based on the theoretical results, we fit an electricity price model for the German intraday market and use it to analyze power cap/floor prices.

Independent increment processes: a multilinearity preserving property

ABSTRACT We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.

Gradual Change Reliability Sensitivity Design for Spindle System on Fully Stochastic Process

Gradual change reliability design of mechanical components is considered by semi-stochastic process model of loading with change of time and fully stochastic process model of known degradation function of strength at present, yet it can’t obtain reasonable reliability design by existing information of components for these models. Thus the influence of existing information and time-dependent parameters of components on reliability should be taken into account in order to access reliability correctly. By studying effect of loading action and time-dependent character of strength, strength of components is taken as a process of independent increments and autocorrelation coefficient of strength is also calculated, meanwhile, we present a method for computing time-dependent reliability. Combining the reliability design theory with sensitivity analysis method, a numerical method for time-dependent reliability sensitivity design of components based on fully stochastic process is proposed, the variation rules of reliability sensitivity of parameters are given at any moment and the effects of design parameters on reliability of components are also studied. The method presented in this paper provides the theoretical basis for structural design and life prediction of mechanical components.

Bayesian degradation assessment of CNC machine tools considering unit non-homogeneity

Field reliability assessment and prediction is critical for the estimation, operation and health management of CNC machine tools. The classical methods for field reliability of CNC Machine Tools assessment and prediction are challenged with the issues of expensive reliability tests, small sample size and unit non-homogeneity. In order to solve these problems, this paper introduces a degradation analysis based reliability assessment method for CNC machine tools under performance testing. Since the degradation is an independent increment process, the gamma process is employed to characterize the degradation process of CNC machine tools. The random effects are introduced to accommodate performance degradation model with unit non-homogeneity. The parameters of model are updated by Bayesian estimation approach. As a case study, the CNC Machine Tools is studied to illustrate the approach. And the proposed method is demonstrated precise for practical use.

Brownian bridge in quantum probability

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L2 space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.

An Improved Independent Increment Process Degradation Model with Bilinear Properties

Degradation analysis can be used to assess the reliability of complex systems and highly reliable products because few or even no failures are expected during their life span. To further previous studies on degradation analysis, an independent increment random process method with linear mean and standard deviation functions is presented to model the degradation procedure. It is essentially a Wiener process method with two different transformed time scales. A one-stage maximum likelihood estimation approach is constructed, and the closed form of the product’s median life and the percentile of the failure time distribution (FTD) are also derived. The proposed method is illustrated and verified in a simulation study and practical degradation analysis for IRLED degradation. The Wiener process model with mixed effects is considered as a reference method. Comparisons show that the difference between the two methods regarding the degradation path is not obvious. However, the estimation accuracy of FTD percentile is significantly enhanced by the proposed model, because a more accurate dispersion can be obtained.

Real-time reliability evaluation based on independent increment process with random effect

Degradation data provide useful information about the reliability assessment for highly reliable and long lifetime products. Motivated by laser degradation data, a new degradation modelling approach is proposed, in which degradation path has linear mean and linear standard deviation functions. In this article, the population degradation modelling and individual real time reliability assessment are discussed, and a Bayesian framework is proposed to integrate population degradation information and individual degradation data. The population degradation path is characterized by a random effect independent increment process where random effect captures unit to unit variation, and the Markov Chain Monte Carlo (MCMC) method is used to estimate the unknown parameters. To obtain individual real time reliability assessment, the parameters are updated iteratively using Bayesian theory. Based on updated results, the residual use life and individual real-time reliability evaluation are obtained. For an illustration of the usefulness and validity of the proposed model and method, a numerical example about laser data is given.

Time sensitive analysis of independent and stationary increment processes

We study the behavior of independent and stationary increments jump processes as they approach fixed thresholds. The exact crossing time is unavailable because the real-time information about successive jumps is unknown. Instead, the underlying process A(t) is observed only upon a third-party independent point process {τn}. The observed time series {A(τn)} presents crude, delayed data. The crossing is first observed upon one of the observations, denoted τν. We develop and further explore a new technique to revive the real-time paths of A(t) for all t belonging to an interval before the pre-crossing observation, [0,τν−1), or between the observations just before and just after the crossing, [τν−1,τν), as a joint Laplace–Stieltjes transform and probability generating function of A(t), A(τν−1), A(τν), τν−1, and τν. Joint probability distributions are obtained from the transforms in a tractable form and they are applied to modeling of stochastic networks under cyber attacks by accurately predicting their crash.

Degradation reliability modeling based on an independent increment process with quadratic variance

Degradation testing is an important technique for assessing life time information of complex systems and highly reliable products. Motivated by fatigue crack growth (FCG) data and our previous study, this paper develops a novel degradation modeling approach, in which degradation is represented by an independent increment process with linear mean and general quadratic variance functions of test time or transformed test time if necessary. Based on the constructed degradation model, closed-form expressions of failure time distribution (FTD) and its percentiles can be straightforwardly derived and calculated. A one-stage method is developed to estimate model parameters and FTD. Simulation studies are conducted to validate the proposed approach, and the results illustrate that the approach can provide reasonable estimates even for small sample size situations. Finally, the method is verified by the FCG data set given as the motivating example, and the results show that it can be considered as an effective degradation modeling approach compared with the multivariate normal model and graphic approach.

On infinitely divisible semimartingales

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.

Reliability analysis of degradation with a new independent increment process

Degradation test is an important method to assess the reliability of complex systems and highly reliable products. The effectiveness of a degradation model depends strongly on the suitability of the model to describe the process. This paper proposes a new degradation model in which the characteristics of the widely used stochastic process and degradation path models are considered simultaneously. According to the proposed model, closed-form expressions of the performance distribution, failure time distribution and their percentiles, as well as reliability, can be obtained easily. A one-stage procedure is then developed to estimate the model parameters, based on which, estimations of the performance distribution, failure time distribution, and reliability are also presented in the paper. Finally, simulation studies are conducted to validate the proposed method. Results suggest that the method provides precise estimates even for zero-failure cases or an extremely small sample size of approximately five.

Futures pricing in electricity markets based on stable CARMA spot models

We present a new model for the electricity spot price dynamics, which is able to capture seasonality, low-frequency dynamics and extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary independent increment process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. The model allows for analytic futures prices, and we apply these to model and estimate the whole market consistently. Besides standard parameter estimation, an estimation procedure is suggested, where we fit the non-stationary trend using futures data with long time until delivery, and a robust L1-filter to find the states of the CARMA process. The procedure also involves the empirical and theoretical risk premia which – as a by-product – are also estimated. We apply this procedure to data from the German electricity exchange EEX, where we split the empirical analysis into base load and peak load prices. We find an overall negative risk premium for the base load futures contracts, except for contracts close to delivery, where a small positive risk premium is detected. Peak load contracts, on the other hand, show a clear positive risk premium, when they are close to delivery, while contracts in the longer end also have a negative premium.

On the oscillation of solutions for a class of second-order nonlinear stochastic difference equations

Abstract In this paper, we investigate the asymptotical behavior for a partial sum sequence of independent random variables, and we derive a law of the iterated logarithm type. It is worth to point out that the partial sum sequence needs not to be an independent increment process. As an application of the theory established, we also give a sufficient criterion on the almost sure oscillation of solutions for a class of second-order stochastic difference equation of neutral type.

Different Random Distributions Research on Logistic-Based Sample Assumption

Logistic-based sample assumption is proposed in this paper, with a research on different random distributions through this system. It provides an assumption system of logistic-based sample, including its sample space structure. Moreover, the influence of different random distributions for inputs has been studied through this logistic-based sample assumption system. In this paper, three different random distributions (normal distribution, uniform distribution, and beta distribution) are used for test. The experimental simulations illustrate the relationship between inputs and outputs under different random distributions. Thereafter, numerical analysis infers that the distribution of outputs depends on that of inputs to some extent, and this assumption system is not independent increment process, but it is quasistationary.

Diffusions of multiplicative cascades

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increment processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers.

Analyzing Degradation by an Independent Increment Process

There is no denying that degradation test is a widely used technique to assess the life‐time information for complex systems and highly reliable products. In this paper, motivated by laser degradation data, valve recession data and independent increment process theory, we propose an independent increment random process method, in which linear mean and standard deviation functions are used to describe the degradation procedure. A one‐stage maximum likelihood estimation of parameters is established. Product's mean‐time‐to‐failure and percentile of the failure time distribution are also derived. The proposed method is illustrated and verified in practice degradation analyzing for the two motivating data sets. It shows that the model can provide reasonable results. Copyright © 2013 John Wiley & Sons, Ltd.

Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions

It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration.