Topic
Reflection principle (Wiener process)
About: Reflection principle (Wiener process) is a research topic. Over the lifetime, 168 publications have been published within this topic receiving 2512 citations.
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TL;DR: In this article, the authors considered the stochastic equation where W(t) is a standard Wiener process and L^X_0 (cdot) is the local time at zero of the unknown process.
Abstract: We consider the stochastic equation $X(t) = W(t) + \beta l^X_0(t)$, where $W$ is a standard Wiener process and $l^X_0(\cdot)$ is the local time at zero of the unknown process $X$. There is a unique solution $X$ (and it is adapted to the fields of $W$) if $|\beta| \leq 1$, but no solutions exist if $|\beta| > 1$. In the former case, setting $\alpha = (\beta + 1)/2$, the unique solution $X$ is distributed as a skew Brownian motion with parameter $\alpha$. This is a diffusion obtained from standard Wiener process by independently altering the signs of the excursions away from zero, each excursion being positive with probability $\alpha$ and negative with probability $1 - \alpha$. Finally, we show that skew Brownian motion is the weak limit (as $n \rightarrow \infty$) of $n^{-1/2}S_{\lbrack nt\rbrack}$, where $S_n$ is a random walk with exceptional behavior at the origin.
333 citations
Book•
01 Jan 1971
TL;DR: The book as discussed by the authors is part of a trilogy covering the field of Markov processes and provides a readable and constructive treatment of Brownian motion and diffusion, which dispenses with most of the customary transform apparatus.
Abstract: : The book is part of a trilogy covering the field of Markov processes and provides a readable and constructive treatment of Brownian motion and diffusion. It contains some of the author's own research and many of the proofs are new. It dispenses with most of the customary transform apparatus, and the chapter on Brownian motion emphasizes topics which haven't had much textbook coverage, such as square variation, the reflection principle, and the invariance principle. The chapter on diffusion shows how to obtain the process from Brownian by changing time. (Author)
287 citations
TL;DR: In this article, it was shown that the moments of a Brownian motion having arbitrary drift and variance and the first time the process dropped a specified amount below its maximum can be found.
Abstract: We determine $E\lbrack \exp (\alpha X(T) - \beta T) \rbrack$ where $X(t)$ is a Brownian motion having arbitrary drift and variance and $T$ is the first time the process drops a specified amount below its maximum to date From this result, the moments of $X(T)$ and $T$ and some asymptotic distributions may be found Applications in process control and financial management are mentioned
151 citations
TL;DR: In this paper, the Erdos-Renyi law of large numbers for the Wiener process was shown to hold for large numbers in large numbers, and connections with strong invariance principles and the P. Levy modulus of continuity for continuity for W(t) were established.
Abstract: Let $\beta_T = (2a_T\lbrack\log(T/a_T) + \log\log T\rbrack)^{-\frac{1}{2}}, 0 0$, the Erdos-Renyi law of large numbers for the Wiener process. A number of other results for the Wiener process also follow via choosing $a_T$ appropriately. Connections with strong invariance principles and the P. Levy modulus of continuity for $W(t)$ are also established.
106 citations
TL;DR: In this paper, the first passage of the integrated Wiener process to 0 was determined in terms of the "$\frac{1}{2}$-winding time" distribution of H. P. McKean, Jr.
Abstract: The rate of first passage of the integrated Wiener process to $x > 0$ is determined in terms of the "$\frac{1}{2}$-winding time" distribution of H. P. McKean, Jr. The probability that the integrated Wiener process is currently at its maximum is approximated.
103 citations