Realization Of Stochastic Process Research Articles

Arma Monte Carlo Simulation in Probabilistic Structural Analysis

Autoregressive moving average (ARMA) systems for sysnthesizing realizations of stochastic processes are discussed in context with the technique of Monte Carlo simulation. Strictly autoregressive (AR) or strictly moving average (MA) systems are considered as special cases of the ARMA systems. Their applicability in wind, ocean, and earthquake engineering is briefly reviewed

The modulator of the local time

AbstractLet x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, αm(x, 1) is integrable over B. The modulator is a function Mm(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the Lm(B)‐valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.

Sojourns and extremes of a stochastic process defined as a random linear combination of arbitrary functions

Let X t be a real stochastic process of the form (X,f(t)) , where X is a random vector in R n with an orthogonally invariant distribution, and f(t),0 ≤ t ≤ 1, assumes values in R n. Put x 0 = sup , finite or infinite. For real u < x 0, let L u be the sojourn time of X t, 0 ≤ t ≤ 1, above u . The main results are limit theorems for the distribution of L u and for P(max t Xt ≥ u), for u ↑ x0 . The hypotheses are stated in terms of the conditions on the tail of the distribution of which are used in extreme value theory for i.i.d. random variables.

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g( x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if | Ee i u′X | ≤ g( u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X( t), 0 ≤ t ≤ 1, to a Gaussian process Y( t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.

Stochastic modeling of mean-wind profiles for in-flight wind estimation--A new approach to lower order stochastic realization schemes

A new approach has been proposed for the design of approximate, lower order discrete time realizations of stochastic processes from the output covariance matrix over a finite time interval. No restrictive assumptions are imposed on the process, that is, it can be nonstationary and also can lead to higher dimension realization. Classes of fixed order models ("guaranteed models") are defined having the joint covariance matrix of the combined vector of the outputs in the interval of definition greater than or equal to the process covariance matrix. The design is achieved by minimizing, in one of those classes, the approximation between the model and the process measured by the trace of the difference of the respective covariance matrices. The method has been employed in a practical problem of developing lower order wind models from a higher order covariance matrix in numerical form. The wind model is intended to be incorporated on board an aircraft maneuvering according to a four-dimensional guidance scheme in the neighborhood of the terminus of a flight. Choice of the order of the model and the resulting accuracies of the estimation schemes are compared.

A Time-Series Analytic Approach to Aggregation Issues in Accounting Data

Since individual elements of corporate financial statements are realizations of stochastic processes, historical accounting records constitute time series and are amenable to time-series analysis. Such analyses may be performed for a variety of purposes, including the development of proxies for market expectations, testing for the significance of accounting changes, the analyses of investment risk as it relates to earnings variability, and considerations of income smoothing. This paper is concerned with temporal aggregation, a characteristic of many accounting time series. This characteristic implies certain timeseries properties and relationships which have not previously been described in the accounting literature. In the next section, a methodology is described which allows the timeseries properties of monthly, quarterly, and annual accounting numbers to be compared analytically. Through the use of several numerical examples, we can assess the possibility of increased estimation efficiency through the use of interim reports, and obtain a measure of the expected relative efficiency of interim data in the forecasting of aggregate data. The analytical results also provide some theoretical support to previous empirical results concerning the time-series properties of accounting numbers.

On some criteria for the realization of stochastic processes

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On some criteria for the realization of stochastic processes

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A Gaussian Paradox: Determinism and Discontinuity of Sample Functions

A real stochastic process $\{X(t): 0 \leqq t \leqq 1\}$, is called window-deterministic if the points $(t, X(t))$ on its graph belonging to a "window" $\{(t, x): 0 \leqq t \leqq 1, a < x < b\}$ stochastically determine all other points on the graph. Here it is shown that a large class of Gaussian processes with discontinuous sample functions has this property.

Exit Properties of Stochastic Processes with Stationary Independent Increments

Let $\{ {X_t},t \geq 0\}$ be a real stochastic process with stationary independent increments. For $x > 0$, define the exit time ${T_x}$ from the interval $( - \infty ,x]$ by ${T_x} = \inf \{ t > 0:{X_t} > x\}$. A reasonably complete solution is given to the problem of deciding precisely when ${P^0}\{ {X_{{T_x}}} = x\} > 0$ and precisely when ${P^0}\{ {X_{{T_x}}} = x\} = 0$. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of ${X_t}$. A few applications of this result are discussed.

Exit properties of stochastic processes with stationary independent increments

Let { X t , t ≥ 0 } \{ {X_t},t \geq 0\} be a real stochastic process with stationary independent increments. For x &gt; 0 x &gt; 0 , define the exit time T x {T_x} from the interval ( − ∞ , x ] ( - \infty ,x] by T x = inf { t &gt; 0 : X t &gt; x } {T_x} = \inf \{ t &gt; 0:{X_t} &gt; x\} . A reasonably complete solution is given to the problem of deciding precisely when P 0 { X T x = x } &gt; 0 {P^0}\{ {X_{{T_x}}} = x\} &gt; 0 and precisely when P 0 { X T x = x } = 0 {P^0}\{ {X_{{T_x}}} = x\} = 0 . The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of X t {X_t} . A few applications of this result are discussed.

On the transformation of the schrödinger wave equation into a system of equations of stochastic processes which have first and second mean square derivatives

It is shown that the Schroedinger wave equation that describes the motion of a particle in a force field derivable from a potential energy can be transformed in a simple way into a system of differential equations for real stochastic processes that have first and second mean square derivative with respect to time.

On the convergence of sequences of stochastic processes

0. Introduction. The purpose of this paper is the formulation and investigation of some convergence concepts for sequences of stochastic processes {xn(t, c), tG [0, 1]}. A related result of these investigations appears as a generalization of the central-limit problem for sequences of sums of independent random variables, embodied in the discussions in ??3 and 5 of the sequence (A); Gnedenko's necessary and sufficient conditions for the convergence in distribution of such sums are used and extended. ?4 contains a version of the Helly-Bray theorem (Theorem 9, Corollary) on probability spaces. The tools used are those developed in [2; 7; 8], and [9] for some special cases of convergence of stable processes. The methods of ?4 are a straightforward adaptation of those of [2]; the convergence property of F [ I used in Theorem 9 was found to be necessary by Udagawa (in the publications of the Union of Japanese Scientists and Engineers) in considering the case of sequences (A) of normed-sum type converging to non-Gaussian stable processes. The processes are throughout this paper assumed to have independent increments. Some special cases of this convergence problem are considered in [1; 3; and 4] without this condition; no general results seem to be known at present. 1. Notation and definitions. Let (Q, 63, p) denote a probability space. A real stochastic process defined on (Q, 63, p) with real parameter set T we shall denote by { x(t, co), tE T}. We shall consider sequences of stochastic processes Xn(t, w), tET}, converging in some sense to a stochastic process {x(t, w), tET}. Let