Penalisation of the Standard Random Walk by a Function of the One-Sided Maximum, of the Local Time, or of the Duration of the Excursions

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.

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On the local time of random walk on the 2-dimensional comb

Stochastic Processes and their Applications ◽

10.1016/j.spa.2011.01.009 ◽

2011 ◽

Vol 121 (6) ◽

pp. 1290-1314 ◽

Cited By ~ 5

Author(s):

Endre Csáki ◽

Miklós Csörgő ◽

Antónia Földes ◽

Pál Révész

Keyword(s):

Random Walk ◽

Local Time

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A Continuous-Time Random Walk Extension of the Gillis Model

Entropy ◽

10.3390/e22121431 ◽

2020 ◽

Vol 22 (12) ◽

pp. 1431

Author(s):

Gaia Pozzoli ◽

Mattia Radice ◽

Manuele Onofri ◽

Roberto Artuso

Keyword(s):

Random Walk ◽

Continuous Time ◽

Continuous Time Random Walk ◽

Hitting Times ◽

Occupation Times ◽

Theoretical Predictions ◽

Normal Diffusion ◽

Heavy Tailed ◽

The One ◽

Waiting Periods

We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.

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Convergence to the local time of Brownian meander

Discrete Mathematics and Applications ◽

10.1515/dma-2019-0014 ◽

2019 ◽

Vol 29 (3) ◽

pp. 149-158 ◽

Cited By ~ 1

Author(s):

Valeriy. I. Afanasyev

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Local Time ◽

Random Processes ◽

Functional Limit Theorem ◽

Zero Drift ◽

Functional Limit ◽

Brownian Meander

Abstract Let {Sn, n ≥ 0} be integer-valued random walk with zero drift and variance σ2. Let ξ(k, n) be number of t ∈ {1, …, n} such that S(t) = k. For the sequence of random processes $\begin{array}{} \xi(\lfloor u\sigma \sqrt{n}\rfloor,n) \end{array}$ considered under conditions S1 > 0, …, Sn > 0 a functional limit theorem on the convergence to the local time of Brownian meander is proved.

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EXTENDED DAILY EXCHANGE RATES FORECASTS USING WAVELET TEMPORAL RESOLUTIONS

New Mathematics and Natural Computation ◽

10.1142/s1793005705000056 ◽

2005 ◽

Vol 01 (01) ◽

pp. 79-107 ◽

Cited By ~ 12

Author(s):

MAK KABOUDAN

Keyword(s):

Neural Networks ◽

Artificial Neural Networks ◽

Exchange Rate ◽

Random Walk ◽

Genetic Programming ◽

Exchange Rates ◽

Us Dollar ◽

Artificial Neural ◽

Japanese Yen ◽

The One

Applying genetic programming and artificial neural networks to raw as well as wavelet-transformed exchange rate data showed that genetic programming may have good extended forecasting abilities. Although it is well known that most predictions of exchange rates using many alternative techniques could not deliver better forecasts than the random walk model, in this paper employing natural computational strategies to forecast three different exchange rates produced two extended forecasts (that go beyond one-step-ahead) that are better than naïve random walk predictions. Sixteen-step-ahead forecasts obtained using genetic programming outperformed the one- and sixteen-step-ahead random walk US dollar/Taiwan dollar exchange rate predictions. Further, sixteen-step-ahead forecasts of the wavelet-transformed US dollar/Japanese Yen exchange rate also using genetic programming outperformed the sixteen-step-ahead random walk predictions of the exchange rate. However, random walk predictions of the US dollar/British pound exchange rate outperformed all forecasts obtained using genetic programming. Random walk predictions of the same three exchange rates employing raw and wavelet-transformed data also outperformed all forecasts obtained using artificial neural networks.

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Zero-one-only process: A correlated random walk with a stochastic ratchet

International Journal of Modern Physics B ◽

10.1142/s0217979214502014 ◽

2014 ◽

Vol 28 (29) ◽

pp. 1450201

Author(s):

Seung Ki Baek ◽

Hawoong Jeong ◽

Seung-Woo Son ◽

Beom Jun Kim

Keyword(s):

Transport Phenomenon ◽

Random Walk ◽

Random Walks ◽

Stochastic Processes ◽

Function Method ◽

Correlated Random Walk ◽

One Dimensional ◽

Persistent Random Walk ◽

The One ◽

Previous Step

The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

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