Partial barrier-absorption probabilities for the Wiener process

1983 ◽  
Vol 20 (1) ◽  
pp. 103-110 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Wiener Process ◽  
Barrier Function ◽  
Standard Wiener Process ◽  
Absorption Probabilities

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The techniques of computing probabilities of the type are well known. The main purpose of this paper is to present ways of finding barrier-absorption probabilities when the barrier function is defined only on sub-intervals of [0, T].

Partial barrier-absorption probabilities for the Wiener process

1983 ◽  
Vol 20 (01) ◽  
pp. 103-110 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Wiener Process ◽  
Barrier Function ◽  
Standard Wiener Process ◽  
Absorption Probabilities

Let {W(t), 0 ≦ t &lt; ∞} be the standard Wiener process. The techniques of computing probabilities of the type are well known. The main purpose of this paper is to present ways of finding barrier-absorption probabilities when the barrier function is defined only on sub-intervals of [0, T].

Evaluations of absorption probabilities for the Wiener process on large intervals

1980 ◽  
Vol 17 (02) ◽  
pp. 363-372 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Stochastic Processes ◽  
Wiener Process ◽  
Standard Wiener Process ◽  
Computationally Efficient ◽  
The Past ◽  
Gaussian Stochastic Processes ◽  
Absorption Probabilities

Let {W(t), 0≦t&lt;∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦T W(t) − f(t) ≧ 0} when either T is large or f(0) &gt; 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

Evaluations of absorption probabilities for the Wiener process on large intervals

1980 ◽  
Vol 17 (2) ◽  
pp. 363-372 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Stochastic Processes ◽  
Wiener Process ◽  
Standard Wiener Process ◽  
Computationally Efficient ◽  
The Past ◽  
Gaussian Stochastic Processes ◽  
Absorption Probabilities

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

Stochastic barriers for the Wiener process

1983 ◽  
Vol 20 (2) ◽  
pp. 338-348 ◽  
Author(s):  
C. Park ◽  
J. A. Beekman
Keyword(s):  
Stochastic Process ◽  
Wiener Process ◽  
Poisson Processes ◽  
Standard Wiener Process ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Deterministic Function

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

scholarly journals A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Fa-mei Zheng
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Wiener Process ◽  
Continuous Distribution ◽  
Random Variables ◽  
Standard Wiener Process ◽  
Trimmed Sums ◽  
Fixed Constant ◽  
Random Products ◽  
Trimmed Sum

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

Evaluations of barrier-crossing probabilities of Wiener paths

1976 ◽  
Vol 13 (02) ◽  
pp. 267-275 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Wiener Process ◽  
Gaussian Processes ◽  
Standard Wiener Process ◽  
Barrier Crossing ◽  
Crossing Probabilities

Let {W(t), 0 ≦ t &lt; ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

scholarly journals THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

2002 ◽  
Vol 18 (1) ◽  
pp. 119-139 ◽  
Author(s):  
Qiying Wang ◽  
Yan-Xia Lin ◽  
Chandra M. Gulati
Keyword(s):  
Wiener Process ◽  
Invariance Principle ◽  
Random Variables ◽  
Linear Process ◽  
Standard Wiener Process ◽  
Test Statistic ◽  
Linear Processes ◽  
Partial Sum ◽  
Unit Root Testing ◽  
Partial Sum Process

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

scholarly journals Optimal control of ultimately bounded stochastic processes

1974 ◽  
Vol 53 ◽  
pp. 157-170
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Markov Process ◽  
Stochastic Differential Equation ◽  
Stochastic Processes ◽  
Wiener Process ◽  
Admissible Control ◽  
Standard Wiener Process ◽  
The Cost

We shall consider the optimal control for a system governed by a stochastic differential equationwhere u(t, x) is an admissible control and W(t) is a standard Wiener process. By an optimal control we mean a control which minimizes the cost and in addition makes the corresponding Markov process stable.

Stochastic barriers for the Wiener process

1983 ◽  
Vol 20 (02) ◽  
pp. 338-348 ◽  
Author(s):  
C. Park ◽  
J. A. Beekman
Keyword(s):  
Stochastic Process ◽  
Wiener Process ◽  
Poisson Processes ◽  
Standard Wiener Process ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Deterministic Function

Let {W(t), 0 ≦ t &lt; ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ T W(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ T W(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

Evaluations of barrier-crossing probabilities of Wiener paths

1976 ◽  
Vol 13 (2) ◽  
pp. 267-275 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Wiener Process ◽  
Gaussian Processes ◽  
Standard Wiener Process ◽  
Barrier Crossing ◽  
Crossing Probabilities

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

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