A General ‘Bang-Bang’ Principle for Predicting the Maximum of a 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.

A General ‘Bang-Bang’ Principle for Predicting the Maximum of a 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.

Coassociative grammar, periodic orbits, and quantum random walk overℤ

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.

A discrete and finite approach to past physical reality

This paper is a synthesis of previously published material on the topic. We show that an adequate mathematical model for the physical (i.e., perceptible and therefore past) reality must be finite. A finite approach to past proper time is given. Proper time turns out to be proportional to the sum of the return probabilities of a Bernoulli random walk.

Searching for a one-dimensional random walker

Let {xk } k ≧ − r be a simple Bernoulli random walk with x –r = 0. An integer valued threshold ϕ = {ϕ k } k≧1 is called a search plan if |ϕ k+1−ϕ k |≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ 1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.

Searching for a one-dimensional random walker

Let {xk}k ≧ − r be a simple Bernoulli random walk with x–r = 0. An integer valued threshold ϕ = {ϕk}k≧1 is called a search plan if |ϕk+1−ϕk|≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.