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The Random Walk Formalism – Bio-Box on Carl Freidrich

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The Random Walk Formalism – Bio-Box on Carl Freidrich 1 1 THE RANDOM WALK A. Chakraborti TOPICS TO BE COVERED IN THIS CHAPTER: What is a Random Walk? • – The random walk formalism – Bio-box on Carl Freidrich Gauss and L Bachelier – The Gaussian distribution – Wiener process – Langevin equation and Brownian motion Do markets follow a random walk (From Bachelier to Eu- • gene Fama & beyond) – “Stylized” facts – ARCH/GARCH processes – Efficient Market Hypothesis (EMH) Power spectral density (PSD) • – Spectral density : Energy and Power – Relation of PSD to auto-correlation – Long-time correlations : Hurst exponent and DFA ex- ponent Econophysics. Sinha, Chatterjee, Chakraborti and Chakrabarti Copyright c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-XXXXX-X 2 1 THE RANDOM WALK 1.1 What is a Random Walk? 1.1.1 Definition of Random walk The mathematical formalization of a trajectory that consists of taking succes- sive “random” (e.g. decided by the flips of an unbiased coin) steps, is known as a random walk. A particularly simple random walk would be that on the integers, which starts at time zero, S = 0 and at each step moves by 1 or 1 with equal prob- 0 − ability (e.g. decided by the flips of an unbiased coin). To define this walk formally, take independent random variables xi, each of which is 1 with prob- ability 1/2 and 1 with probability 1/2, and set S = Σn x .This sequence − n i=1 i Sn is called the simple random walk on integers. This walk can be illustrated (see 1.1) as follows: Say you flip an unbiased coin. If it lands on heads H, you move one to the right on the number line, and if it lands on tails T, then you move one to the left. So after five flips, you have the possibility of landing on 1, 1,3, 3,5, 5. You can land on 1 by − − − flipping three heads and two tails in any order. There are 10 possible ways of landing on 1. Similarly, there are 10 ways of landing on -1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on -3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on -5 (by flipping five tails). These results are directly related to the properties of Pascal’s triangle. The number of different walks of n steps where each step is +1 or -1 is clearly 2n. For the simple random walk, each of these walks are equally likely. In order for Sn to be equal to a number k, it is necessary and sufficient that the number of +1 in the walk exceeds those of -1 by k. Thus, the number of walks which satisfy Sn = k is precisely the number of ways of choosing (n + k)/2 elements from an n element set (for this to be non-zero, it is necessary that n + k be an even number), which is an entry in Pascal’s triangle denoted by n n n C(n+k)/2. Therefore, the probability that Sn = k is equal to 2− C(n+k)/2. This relation with Pascal’s triangle (see 1.2) is easily demonstrated for small values of n. At zero turns, the only possibility will be to remain at zero. How- ever, at one turn, you can move either to the left or the right of zero, meaning there is one chance of landing on -1 or one chance of landing on 1. At two turns, you examine the turns from before. If you had been at 1, you could move to 2 or back to zero. If you had been at -1, you could move to -2 or back to zero. So there is one chance of landing on -2, two chances of landing on zero, and one chance of landing on 2. We shall study more interesting aspects of the random walk later in this chapter. 1.1 What is a Random Walk? 3 Table 1.1 Random coin flips. If there is a head H we move right on the number line (add +1), and if there is a tail T we move left on the number line (add -1). Table 1.2 Pascal’s triangle. The results of random walk analysis is central in physics, chemistry, eco- nomics and a number of other fields as a fundamental model for random (stochastic) processes in time. There are many systems for which at smaller scales, the interactions with the environment and their influence are in the form of random fluctuations, as in the case of “Brownian motion” 1. If the mo- 1) The motion of the particle is called Brownian Motion, in honor to the botanist Robert Brown who observed it for the first time in his 4 1 THE RANDOM WALK 30 20 10 0 y −10 −20 −30 −40 −50 −100 −80 −60 −40 −20 0 20 x Table 1.3 Simulated Brownian motion (5000 time steps). tion of a pollen grain in a fluid like water is observed under a microscope, it would look somewhat like what is shown in the figure 1.3. It is interesting to note that the path traced by the pollen grain as it travels in a liquid (observed by R. Brown and studied first by A. Einstein), and the price of a fluctuating stock (studied first by L. Bachelier), can both be modeled as random walks (theory of stochastic processes). It is noteworthy that the formulation of the random walk model — as well as of a stochastic process — was first done in the framework of the economic study by L. Bachelier [31, 32], even five years prior to the work of A. Einstein! There are of course other systems, that present unpredictable “chaotic” be- havior, this time due to dynamically generated internal “noise”. Noisy pro- cesses in general, either truly stochastic or chaotic in nature, represent the rule rather than the exception. In this chapter, we will concentrate only on the former theory of random or stochastic processes. ******************************************************************************* BIO- BOX ON JOHANN CARL FRIEDRICH GAUSS (from wikipedia) Johann Carl Friedrich Gauss (30 April 1777 âA¸S23˘ February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential ge- ometry, geodesy, geophysics, electrostatics, astronomy and optics. Some- times known as the Princeps mathematicorum (the Prince of Mathemati- cians) greatest mathematician since antiquity, Gauss had a remarkable in- fluence in many fields of mathematics and science and is ranked as one of history’s most influential mathematicians. studies of pollen. In 1828 he wrote “the pollen become dispersed in water in a great number of small particles which were perceived to have an irregular swarming motion”. The theory of such motion, however, was derived by A. Einstein in 1905 when he wrote: “In this paper it will be shown that ... bodies of microscopically visible size suspended in a liquid perform movements of such magnitude that they can be easily observed in a microscope on account of the molecular motions of heat ...” 1.1 What is a Random Walk? 5 Gauss was born on April 30, 1777 in Braunschweig, in the Electorate of Brunswick-LÃijneburg, now part of Lower Saxony, Germany, as the sec- ond son of poor working-class parents. There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances. Gauss attended the Collegium Carolinum (now Technische Universität Braunschweig from 1792 to 1795, and subsequently he moved to the Uni- versity of Göttingen from 1795 to 1798. His breakthrough occurred in 1796 when he was able to show that any regular polygon with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. This was a major discovery in an important field of mathematics; construction prob- lems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. In his 1799 doctorate in absentia, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polyno- mial over the complex numbers has at least one root. Gauss also made im- portant contributions to number theory with his 1801 book Disquisitiones Arithmeticae, which contained a clear presentation of modular arithmetic and the first proof of the law of quadratic reciprocity. In that same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres, but could only watch it for a few days. Gauss predicted cor- rectly the position at which it could be found again, and it was rediscov- ered by Franz Xaver von Zach on 31 December 1801 in Gotha, and one day later by Heinrich Olbers in Bremen. In 1807, Gauss was appointed Professor of Astronomy and Director of the astronomical observatory in GÃ˝uttingen, a post he held for the remainder of his life. The discovery of Ceres by Piazzi on 1 January 1801 led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventu- ally published in 1809. It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of mea- surement error.
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