Elements of Stochastic Processes

A stochastic process in the time interval [0,∞) is defined as a family of random variables X t (ω), t ≥ 0, ω ∈ Ω, on a measurable space (Ω,F, P). To describe the probability structure of the process, one should define the family of distributions of (Xt1,…, Xtn) for any integer n and any 0 ≤ t 1 < t 2 < … < tn. This family of finite dimensional distributions is, in general, a rather bulky subject. Therefore, as a rule, special classes of stochastic processes are considered for which such a description can be reduced to more convenient characteristics. First, we consider the basic processes which have a simple statistical structure. Then we consider relatively simple and explicitly defined transformations of these basic processes. The Wiener process, Poisson process, and continuous time Markov chains with finite number of states will be our basic processes. We assume that the main properties of these processes are known. Actually, all classes of continuous time stochastic processes, allowing deep enough theory, can be constructed from these basic processes using relatively simple transformations.

Chapter Four - STOCHASTIC PROCESSES

Generating Functions Recurrent Events Random Walk Models Markov Chains Discrete Branching Processes Markov Processes in Continuous Time Homogeneous Birth and Death Processes Some Non-Homogeneous Processes Multi-Dimensional Processes Queueing Processes Epidemic Processes Competition and Predation Diffusion Processes Approximations to Stochastic Processes Some Non-Markovian Processes.

Suppose {X(t), t ∈ G d } is a random field in d dimensions, with d ∈ ℤ+; that is, {X(t), t ∈ G d } is a collection of random variables X(t) taking values in a state space S, defined on a probability space (Ω, A, P), and indexed by the variable t ∈ G d . Throughout this chapter, G will stand for either the set of real numbers ℝ, or the set of integers ℤ; thus, the random field {X(t)} is allowed to “run” in either continuous or discrete “time.” Similarly, G+ will denote ℝ+ or ℤ+ (the sets of positive real numbers and integers, respectively) according to whether G = ℝ or G = ℤ.

For autoregressive moving average (ARMA) models with infinite variance innovations, quasi-likelihood-based estimators (such as Whittle estimators) suffer from complex asymptotic distributions depending on unknown tail indices. This makes statistical inference for such models difficult. In contrast, the least absolute deviations estimators (LADE) are more appealing in dealing with heavy tailed processes. In this paper, we propose a weighted least absolute deviations estimator (WLADE) for ARMA models. We show that the proposed WLADE is asymptotically normal, is unbiased, and has the standard root-n convergence rate even when the variance of innovations is infinity. This paves the way for statistical inference based on asymptotic normality for heavy-tailed ARMA processes. For relatively small samples numerical results illustrate that the WLADE with appropriate weight is more accurate than the Whittle estimator, the quasi-maximum-likelihood estimator (QMLE), and the Gauss–Newton estimator when the innovation variance is infinite and that the efficiency loss due to the use of weights in estimation is not substantial.The authors thank the two referees for their valuable suggestions. The work was partially supported by an EPSRC research grant (UK) and the Natural Science Foundation of China (grant 10471005).

Part VI - Elements of Stochastic Processes

This book has its origin in the need for developing and analyzing mathematical models for phenomena that evolve in time and influence each another, and aims at a better understanding of the structure and asymptotic behavior of stochastic processes. This monograph has double scope. First, to present tools for dealing with dependent structures directed toward obtaining normal approximations. Second, to apply the normal approximations presented in the book to various examples. The main tools consist of inequalities for dependent sequences of random variables, leading to limit theorems, including the functional central limit theorem (CLT) and functional moderate deviation principle (MDP). The results will point out large classes of dependent random variables which satisfy invariance principles, making possible the statistical study of data coming from stochastic processes both with short and long memory. Over the course of the book different types of dependence structures are considered, ranging from the traditional mixing structures to martingale-like structures and to weakly negatively dependent structures, which link the notion of mixing to the notions of association and negative dependence. Several applications have been carefully selected to exhibit the importance of the theoretical results. They include random walks in random scenery and determinantal processes. In addition, due to their importance in analyzing new data in economics, linear processes with dependent innovations will also be considered and analyzed.

Stochastic processes are expressive mathematical tools for modeling real-world systems that are subject to uncertainty. It is hence crucial to be able to formally analyze the behavior of these processes, especially in safety-critical applications. Most of the existing formal methods are not designed for continuous-time processes, and those that are typically suffering from state explosion in practice. This article introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The framework is demonstrated in three case studies, including one that illustrates the tradeoff of the error contribution by the time and space discretization parameters.

The Elements of Stochastic Processes with Applications to the Natural Sciences

The Elements of Stochastic Processes with Application to the Natural Sciences

In probability theory, a family $$ \left\{ {x_t ,t \in I} \right\} $$ of random variables on a probability space $$ \left( {S,\mathbb{S}\prime ,P} \right), $$ that is, a family of measurable functions from S into $$ \bar R $$ is sometimes glorified by the name stochastic process or, simply, process. At the point s of S, the random variable x t has value x t (s), a point of the state space of the process, and the function x•(s), from I into the state space, is a sample function of the process. The set I is the parameter set of the family. There is no definition of the term stochastic process more specific than that just given, but the term is usually reserved for families with some interesting property. To loosely paraphrase a judge discussing a somewhat different concept, probabilists cannot define stochastic processes, but they recognize one when they see it! Typical examples of the application of the term are to martingales and to Markov processes, both defined in this chapter.KeywordsConditional ExpectationMeasure TheoryHarmonic MeasureInfinite SequenceIndex PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter is devoted to further topics in the theory of stochastic processes and their applications. We start with a weaker definition of a stochastic process that is sufficient in the study of stationary processes. We said before that a stochastic process is a function u of both a variable ω in a probability space and a continuous parameter t, making u a random variable for each t and a function of t for each ω. We made statements about the kind of function of t that was obtained for each ω. The definition here is less specific about what happens for each ω.KeywordsStochastic ProcessCharacteristic FunctionCovariance FunctionRandom MeasureWide SenseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The problem of obtaining all stochastic realizations of a discrete-time non stalionary stochastic process defined on the set of integers is considered. The structure of the set of all covariances of the states of such realizations is studied. Special attention is paid to the minimum and maximum elements of this set. Finally, a non-Riccati-type algorithm—which does not require solution of a matrix Riccati equation—to generate (wide-sense) stochastic realizations is presented.

Generally, the likelihood ratio statistic \(\varLambda \) for standard hypotheses is asymptotically \(\chi ^2\) distributed, and the Bartlett adjustment improves the \(\chi ^2\) approximation to its asymptotic distribution in the sense of third-order asymptotics. However, if the parameter of interest is on the boundary of the parameter space, Self and Liang (1987) show that the limiting distribution of \(\varLambda \) is a mixture of \(\chi ^2\) distributions. For such “nonstandard setting of hypotheses”, the current chapter develops the third-order asymptotic theory for a class \(\mathcal {S}\) of test statistics, which includes the Likelihood Ratio, the Wald and the Score statistic, in the case of observations generated from a general stochastic process, providing widely applicable results. In particular, it is shown that \(\varLambda \) is Bartlett adjustable despite its nonstandard asymptotic distribution. Although the other statistics are not Bartlett adjustable, a nonlinear adjustment is provided for them which greatly improves the \(\chi ^2\) approximation to their distribution and allows a subsequent Bartlett type adjustment. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to \(\mathcal {S}\).

Fractional Brownian Motions, Fractional Noises and Applications