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Top1. Introduction
The understanding of the underlying behaviour of financial markets is of fundamental importance. Although financial markets are very diverse, they share many similarities to complex models traditionally studied by physicists. Many phenomena found and studied in statistical physics can be found to also occur in various financial markets (equities, currencies or commodities). Whereas model systems have been extensively studied over the centuries in physics, it’s only over the past two decades that the ideas and tools used in physics have been applied to financial markets. The objective is to try and understand the underlying behaviour of financial markets. The application of physics to the fields of finance and economic has led to the new discipline of “Econophysics.” The main motivation of the present work is to study data resulting from the Japanese stock market using a recently discovered result in statistical physics, namely that of “persistence”.
In its most generic form, the so-called “persistence” problem in physics is concerned with the fraction of space which literally persists in its initial (t = 0) state up to some later time t. It is a classic problem which falls into the general class of so-called “first passage” problems (Redner, 2001) and has been extensively studied over the past two decades or so for model spin systems by physicists (Bray, Derrida, & Godreche, 1994; Derrida, Bray, & Godreche, 1994; Stauffer, 1994; Derrida, Hakim, & Pasquier, 1995, 1996; Majumdar, Bray, Cornell, & Sire, 1996). Persistence has been investigated at both zero (Bray et al., 1994; Derrida et al., 1994; Stauffer, 1994; Derrida, et al., 1995, 1996) and non-zero (Majumdar et al., 1996) temperatures.
Typically, in the non-equilibrium dynamics of spin systems at zero-temperature (Bray et al., 1994; Derrida et al., 1994; Stauffer, 1994; Derrida, et al., 1995, 1996), the system is prepared initially in a random state and the fraction of spins, P(t), that persists in the same state as at t = 0 up to some later time t is monitored. At a finite temperature, on the other hand, one is interested in the global persistence behaviour and one monitors the change in the sign of the magnetization in a collection of non-interacting systems (Majumdar et al., 1996). It is now well established that the persistence probability decays algebraically (Bray et al., 1994; Derrida et al., 1994; Stauffer, 1994; Derrida, et al., 1995, 1996; Majumdar et al., 1996):
(1) where θ(d, q) is a new non-trivial persistence exponent. Note that the value of θ depends not only on the spatial (Stauffer, 1994) (d) and the spin (Derrida, de Oliveira, & Stauffer, 1996) (q) dimensionalities, but also on whether the temperature, T, is zero or finite. It is only for T = 0 and d = 1 that θ (1, q) is known exactly (Derrida et al., 1995, 1996); see Ray (2004) for a review. We merely mention here that at criticality, T = T
c, θ (2, 2) ∼ 0.5 for the pure two-dimensional Ising model (Majumdar et al., 1996).