Asymptotic expansions for the distribution of the number of incidences in symmetric Bernoulli random walk segments

1974 ◽  
Vol 14 (4) ◽  
pp. 565-572
Author(s):  
A. T. Semenov
Keyword(s):  
Random Walk ◽  
Asymptotic Expansions ◽  
Bernoulli Random Walk

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (04) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T E[f(M T − B τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B t ) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B t ) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (4) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

scholarly journals Coassociative grammar, periodic orbits, and quantum random walk overℤ

2005 ◽  
Vol 2005 (24) ◽  
pp. 3979-3996 ◽  
Author(s):  
Philippe Leroux
Keyword(s):  
Random Walk ◽  
Periodic Orbits ◽  
Chaotic System ◽  
Quantum Random Walk ◽  
Bernoulli Random Walk

Inspired by a work of Joni and Rota, we show that the combinatorics generated by a quantisation of the Bernoulli random walk overℤcan be described from a coassociative coalgebra. Relationships between this coalgebra and the set of periodic orbits of the classical chaotic systemx↦2x mod⁡1,x∈[0,1], are also given.

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