Random walk hypothesis

Summary

The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted. History The concept can be traced to French broker Jules Regnault who published a book in 1863, and then to French mathematician Louis Bachelier whose Ph.D. dissertation titled "The Theory of Speculation" (1900) included some remarkable insights and commentary. The same ideas were later developed by MIT Sloan School of Management professor Paul Cootner in his 1964 book The Random Character of Stock Market Prices. The term was popularized by the 1973 book A Random Walk Down Wall Street by Burton Malkiel, a professor of economics at Princeton University, and was used earlier in Eugene Fama's 1965 article "Random Walks In Stock Market Prices", which was a less technical version of his Ph.D. thesis. The theory that stock prices move randomly was earlier proposed by Maurice Kendall in his 1953 paper, The An

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https://en.wikipedia.org/wiki/Random_walk_hypothesis

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Though the following topics seem unlinked, most of the tools used in this thesis are related to random walks and renewal theory. After introducing the voter model, we consider the parabolic Anderson model with the voter model as catalyst. In GÄRTNER, DEN HOLLANDER and MAILLARD [44], the behaviour of the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant with respect to the catalyst, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In Chapter 3 we address some questions left open in this paper by considering specifically when the Lyapunov exponents are the a priori maximal value. Then, we use exclusion process techniques to show that the evolution of a perturbed threshold voter model is recurrent in the critical case. The key to our approach is to develop the ideas of BRAMSON and MOUNTFORD [9] : we exhibit a Lyapunov-Foster function for the discrete time version of the process. We also make a widespread use of coupling arguments. Finally, using the regenerative scheme of COMETS, FERNÁNDEZ and FERRARI [19], we establish a functional central limit theorem for discrete time stochastic processes with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the limit theorem is fulfilled.

EPFL2012

Interfaces of the one-dimensional voter models

Samir Brahim Belhaouari

We show that for a voter model on {0,1}Z corresponding to a random walk with kernel p(·) and starting from unanimity to the right and opposing unanimity to the left, a tight interface between 0's and 1's exists if p(·) has second moments but does not if p(·) fails to have α moment for some α < 2. We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite γth moment for some γ > 3, then the evolution of the interface boundaries converges weakly to a Brownian motion. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that a finite γth moment is necessary for this convergence for all γ ∈ (0, 3). We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett.

EPFL2006

Metastability of reversible condensed zero range processes on a finite set

Johel Beltran Ramirez

Let r : S x S -> R+ be the jump rates of an irreducible random walk on a finite set S, reversible with respect to some probability measure m. For alpha > 1, let g : N -> R+ be given by g(0) = 0, g(1) = 1, g(k) = (k/k - 1)(alpha), k >= 2. Consider a zero range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). Let N stand for the total number of particles. In the stationary state, as N up arrow infinity, all particles but a finite number accumulate on one single site. We show in this article that in the time scale N1+alpha the site which concentrates almost all particles evolves as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.

2012