Independent Increments Research Articles

Characterization of a wiener process taking values in a Hilbert space

This paper characterize wiener process by taking values in a Hilbert space. A standard wiener process is stochastic process indesed by nonnegative real numbers t with the following properties: W0=0 With probability 1, the function t àWt is continuous in t. The process has stationary, independent increments. The increments - Ws has the NORMAL (0,t) distribution

Integrating age modeling into a hierarchical Bayesian framework for inferring the pattern and rate of past sea-level changes from uncertainty-prone proxy data

Inferring the pattern and rate of past sea-level changes from uncertainty-prone proxy records requires formal statistical analyses, preferably in a hierarchical framework. The commonly used error-in-variables method treats the relative sea level as a collection of random variables drawn from the multivariate Gaussian distribution. However, this method does not make any use of prior information about the sea-level index points as constraints in the inferential process, thereby leading to anomalously large uncertainties for the time periods when observational data are absent. Here, a hierarchical Bayesian model of past sea-level changes is presented. Specifically, the stochastically varying relative sea level is modeled as a piecewise linear process with an additive independent Brownian increment arriving in a Gaussian fashion. The treatment of temporal uncertainties associated with the sea-level index points in the partially observed proxy records also differs from the existing methods. Instead of calibrating the radiocarbon ages individually, the corresponding calendar ages are treated as random variables and inferred recursively according to their temporal order. Illustrative studies using synthetic and real-world data demonstrate the promise of this model.

A cumulative damage model for tubes in nuclear power plants subject to continuous degradation and multi-effect shocks under dynamic environments

Continuous degradation and shocks are two vital damage mechanisms for tubes under dynamic environments in nuclear power plants (NPPs). In this work, we presented a cumulative damage model to evaluate the reliability of tubes in NPPs, which accounts for continuous degradation and random shocks. The dynamic environments were assumed to obey a homogenous Markov process. Continuous degradation was described as a general stochastic process with independent increments. A multinomial distribution was used to depict the multiple effects of Poisson-distributed shocks on tubes. The influence of dynamic environments on the cumulative damage process of tubes was also explicitly modeled. Closed-form reliability measures were derived, such as reliability functions and mean time to failure (MTTF) of tubes. Drawing upon a case study from the existing literature, the analytical solution was validated through a comparative analysis with the outcomes of Monte Carlo simulation (MCS) and other methodologies referenced in the literature. As a practical engineering application, the reliability of steam generator tubes made of Alloy 690 within pressurized water reactors (PWRs), which are susceptible to primary side stress corrosion cracking (PWSCC) and the effects of water hammer, was calculated. The findings indicate that the probabilities of steam generator tube rupture (SGTR) as estimated by the model and those inferred from operational experience have the same order of magnitude.

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.

Limiting Ramp Rate of Wind Power Output using a Battery Based on the Variance Gamma Process

A battery system is designed to reduce violations of ramping limit due to severe and long ramps of wind power output. The power rating, battery capacity, and operational policies of the battery are decided based on the assumption that the wind power output follows the Variance Gamma (V.G.) process over short time interval. The histogram of the ramp rates of wind power follows the symmetric Laplace distribution. Furthermore, ramp rates have independent and stationary increments, and their variances are proportional to the time interval. Consequently, the wind power output can be modelled as a simple form of the V.G. process. Battery operational oplicies based on the V.G. process are tested using statistics representing the sum of all wind power outputs from wind farms in the Electric Reliability Council of Texas (ERCOT). It is found that the battery can essentially eliminate violations of ramping limits due to severe ramp events. Furthermore, in this operation, it is discovered that the power rating influences the ramp limiting performance of the battery more than battery storage rating.

Large Sample Asymptotic Analysis for Normalized Random Measures with Independent Increments

Large Sample Asymptotic Analysis for Normalized Random Measures with Independent Increments

Group sequential two-stage preference designs.

The two-stage preference design (TSPD) enables inference for treatment efficacy while allowing for incorporation of patient preference to treatment. It can provide unbiased estimates for selection and preference effects, where a selection effect occurs when patients who prefer one treatment respond differently than those who prefer another, and a preference effect is the difference in response caused by an interaction between the patient's preference and the actual treatment they receive. One potential barrier to adopting TSPD in practice, however, is the relatively large sample size required to estimate selection and preference effects with sufficient power. To address this concern, we propose a group sequential two-stage preference design (GS-TSPD), which combines TSPD with sequential monitoring for early stopping. In the GS-TSPD, pre-planned sequential monitoring allows investigators to conduct repeated hypothesis tests on accumulated data prior to full enrollment to assess study eligibility for early trial termination without inflating type I error rates. Thus, the procedure allows investigators to terminate the study when there is sufficient evidence of treatment, selection, or preference effects during an interim analysis, thereby reducing the design resource in expectation. To formalize such a procedure, we verify the independent increments assumption for testing the selection and preference effects and apply group sequential stopping boundaries from the approximate sequential density functions. Simulations are then conducted to investigate the operating characteristics of our proposed GS-TSPD compared to the traditional TSPD. We demonstrate the applicability of the design using a study of Hepatitis C treatment modality.

Electricity consumption simulation using random coefficient periodic autoregressive model

The concept of a continuous-time conditional linear random process involves a random kernel being integrated stochastically by a process with independent increments, which is often called a “generative process”. In cases where the “generative process” is Poisson, the resulting model represents an investigated signal as a sum of numerous stochastically dependent random impulses each of which occurs according to some inhomogeneous Poisson arrival process. This model can be applied to represent various processes related to energy consumption, such as electricity loads of electrical power systems, gas and water consumption, and other energy resources, while also considering the signals’ cyclostationarity, which is usually caused by the rhythmic nature of consumer behaviour. A member of the discretetime conditional linear cyclostationary random processes class is the random coefficient periodic autoregressive (RCPAR) model, which is appropriate for use in energy informatics, including estimation, forecasting, and computer simulation purposes. The primary objective of the paper is to establish a procedure for simulating the hourly electricity consumption of small and medium-sized enterprises using the RCPAR model, which has periodic parameters and creates cyclostationary properties while also accounting for the investigated process conditional heteroscedasticity. The statistical estimation step of the proposed procedure uses the general methodology for estimating the parameters of the RCPAR model and the methods of statistical analysis of cyclostationary signals. This step is used to identify the simulation characteristics. The simulation step is based on the methods of cyclostationary white noise generation and its transformation by a digital linear filter with random parameters. The last ones are obtained using the Gaussian random vectors computer simulation methods, taking into account the cyclostaionarity property.

A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes

We establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.

On the local time of Gaussian and Lévy processes

Abstract The local time (LT) of a given stochastic process { X t : t ≥ 0 } \{X_{t}:t\geq 0\} is defined informally as L X ⁢ ( t , x ) = ∫ 0 t δ x ⁢ ( X s ) ⁢ d s , L_{X}(t,x)=\int_{0}^{t}\delta_{x}(X_{s})\,ds, where δ x \delta_{x} denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ⁢ ( u ) ⁢ d B u X_{t}:=\int_{0}^{t}f(u)\,dB_{u} for a given deterministic function f ∈ L 2 ( [ 0 , + ∞ [ ) f\in L^{2}([0,+\infty[) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.

Simplified calculus for semimartingales: Multiplicative compensators and changes of measure

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation when such compensation is meaningful. This generalization of the Lévy–Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a signed stochastic exponential, which in turn has practical applications in mean–variance portfolio theory. Girsanov-type results based on multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the characteristic function of log returns for a popular class of minimax measures in a Lévy setting.

Lévy models for collapse of the wave function

Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.

On Quasi-Infinitely Divisible Random Measures

Quasi-infinitely divisible (QID) distributions have been recently introduced. A random variable X is QID if and only if there exist two infinitely divisible random variables Y and Z s.t. X+Y=dZ and Y is independent of X. In this work, we introduce QID completely random measures (CRMs) and show that a certain family of QID CRMs is dense in the space of all CRMs with respect to convergence in distribution. We further demonstrate that the elements of this family possess a Lévy-Khintchine formulation and that there exists a one to one correspondence between their law and certain characteristic pairs. We prove the same results also for the class of point processes with independent increments. In the second part of the paper, we show the relevance of these results in a general Bayesian nonparametric framework suitable for topic modelling, and provide a truncation error analysis.

Average preserving variation processes in view of optimization

In this paper, we investigate specific least action principles for laws of stochastic processes within a framework which stands on filtrations preserving variations. The associated Euler–Lagrange conditions, which we obtain, exhibit a deterministic process in the dynamics aside the canonical martingale term. In particular, taking specific action functionals, extremal processes with respect to those variations encompass specific laws of continuous semi-martingales whose drift characteristic is integrable with independent increments. Then, we relate extremal processes of classical cost functions, in particular of specific entropy functions, to a class of forward-backward systems of Mckean–Vlasov stochastic differential equations.

Semiparametric predictive inference for failure data using first-hitting-time threshold regression.

The progression of disease for an individual can be described mathematically as a stochastic process. The individual experiences a failure event when the disease path first reaches or crosses a critical disease level. This happening defines a failure event and a first hitting time or time-to-event, both of which are important in medical contexts. When the context involves explanatory variables then there is usually an interest in incorporating regression structures into the analysis and the methodology known as threshold regression comes into play. To date, most applications of threshold regression have been based on parametric families of stochastic processes. This paper presents a semiparametric form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this model has potential for use in many fields. The mathematical underpinnings of this semiparametric approach for estimation and prediction are described. The basic data element required by the model is a pair of readings representing the observed change in time and the observed change in disease level, arising from either a failure event or survival of the individual to the end of the data record. An extension is presented for applications where the underlying disease process is unobservable but component covariate processes are available to construct a surrogate disease process. Threshold regression, used in combination with a data technique called Markov decomposition, allows the methods to handle longitudinal time-to-event data by uncoupling a longitudinal record into a sequence of single records. Computational aspects of the methods are straightforward. An array of simulation experiments that verify computational feasibility and statistical inference are reported in an online supplement. Case applications based on longitudinal observational data from The Osteoarthritis Initiative (OAI) study are presented to demonstrate the methodology and its practical use.

Finite-dimensional Discrete Random Structures and Bayesian Clustering

Discrete random probability measures stand out as effective tools for Bayesian clustering. The investigation in the area has been very lively, with a strong emphasis on nonparametric procedures based on either the Dirichlet process or on more flexible generalizations, such as the normalized random measures with independent increments (NRMI). The literature on finite-dimensional discrete priors is much more limited and mostly confined to the standard Dirichlet-multinomial model. While such a specification may be attractive due to conjugacy, it suffers from considerable limitations when it comes to addressing clustering problems. In order to overcome these, we introduce a novel class of priors that arise as the hierarchical compositions of finite-dimensional random discrete structures. Despite the analytical hurdles such a construction entails, we are able to characterize the induced random partition and determine explicit expressions of the associated urn scheme and of the posterior distribution. A detailed comparison with (infinite-dimensional) NRMIs is also provided: indeed, informative bounds for the discrepancy between the partition laws are obtained. Finally, the performance of our proposal over existing methods is assessed on a real application where we study a publicly available dataset from the Italian education system comprising the scores of a mandatory nationwide test.

Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions

In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although the two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

Simulation of the nonlinear Kerr and Raman effect with a parallel local time-stepping DGTD solver.

In this paper, an efficient discontinuous Galerkin time-domain (DGTD) method is proposed to solve Maxwell's equations for nonlinear Kerr or Raman media. Based on our previous work, an arbitrary high-order derivatives DGTD method with a local time-stepping scheme is introduced for simulating dynamic optical responses in nonlinear dispersive media such that the nonlinear effects do not impose constraints on the stability conditions for linear subdomains. Therefore, the scheme enables the simulations in the nonlinear and linear media regions with independent time-stepping increments, which greatly improves the efficiency of the time-domain analysis. Moreover, by applying an iteration solution scheme, the proposed method preserves the intrinsic local features, which is favorable for the realization of highly parallelized algorithms. Numerical examples demonstrate the accuracy and the efficiency of our proposed method. We believe the proposed method provides an effective tool for numerical analysis of nonlinear optical phenomena.

Epidemics from the Eye of the Pathogen

While a common trend in disease modeling is to develop models of increasing complexity, it was recently pointed out that outbreaks appear remarkably simple when viewed in the incidence vs. cumulative cases (ICC) plane. This article details the theory behind this phenomenon by analyzing the stochastic SIR (Susceptible, Infected, Recovered) model in the cumulative cases domain. We prove that the Markov chain associated with this model reduces, in the ICC plane, to a pure birth chain for the cumulative number of cases, whose limit leads to an independent increments Gaussian process that fluctuates about a deterministic ICC curve. We calculate the associated variance and quantify the additional variability due to estimating incidence over a finite period of time. We also illustrate the universality brought forth by the ICC concept on real-world data for Influenza A and for the COVID-19 outbreak in Arizona.

Non-commutative stochastic processes with independent increments

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels.