The traditional efficiency market hypothesis supposes that the fluctuations of the stock prices are random, and stock price is unable to be predicted. But in recent years papers point out that the fluctuations of the stock prices are not totally random, the fluctuations of the stock price have long term memory characteristics. Therefore, trying to find out the regularity of the market price becomes a new subject for research. This paper attempts to use the fractional market hypothesis to analyze stock market, which divided samples into two types which are the developed markets ¡]Japan , U.S.A. , Australia, and South Africa¡^and mergering markets ¡]Korea , Taiwan , China¡]Shanghai¡^ , and Jordan¡^. The sample period is from January of 1997 to December of 2006. And using the regarding countries¡¦ main returns of daily stock price index. By using R/S analysis to estimate each country¡¦ Hurst coefficients, this paper studies the aperiodic cycle in each country. It also wants to see whether the degree of maturity affects the different result or not. The empirical results show that the stock indeies in the developed markets have shorter aperiodic cycle than in the mergering markets. U.S.A., Australia ,and South African markets the aperiodic cycles are 138 days , 126 days ,and 152 days respectively. Taiwan and Shanghai markets the aperiodic cycles are 208 days and 202 days respectively. Japan, Korea , Jordanian markets in this sample period have not found aperiodic cycles.

The main goal of this thesis is to test the ability of the Hurst exponent to recognise some processes with deterministic signal as nonrandom and to test the randomness of daily stock returns of three stocks traded in BCPP. Critical values to determine the critical region of a randomness hypothesis test were set for this purpose. Another goal of the thesis is the description of the Hurst exponent estimation by means of Rescaled Range Analysis and outline some problems accompanying this estimation if the Hurst exponent would be used as a randomness indicator. Within the frame of Rescaled Range Analysis was constructed another method that showed to be successful in recognising some series that contain deterministic signal.

A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice. / In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.

El presente trabajo comprueba la hipótesis de eficiencia de mercado (EMH) a través de una medida de persistencia temporal conocida como exponente de Hurst. Esta aproximación además de estar relacionada con la dimensión fractal, permite expandir el análisis de la hipótesis de mercado eficiente, propuesta por Eugene Fama en 1970. El cálculo del exponente de Hurst se realiza en base al método de rango reescalado; y se extiende su aplicación a una estimación dinámica entre el periodo 2006-2021. Este indicador sirve como índice de eficiencia de mercado, y se estima para las series de retornos diarios de los mercados de valores del MILA, conformado por Chile, Colombia, México y Perú. Los resultados demuestran que Perú es el mercado menos eficiente, y con mayor número de ciclos de ineficiencia para el periodo calculado. Por otro lado, México resulta ser el único mercado del MILA que estuvo dentro de la región de eficiencia. / The present work tests the market efficiency hypothesis (EMH) through a measure of time persistence known as the Hurst exponent. This approach, in addition to being related to the fractal dimension, allows us to expand the analysis of the efficient market hypothesis proposed by Eugene Fama in 1970. The calculation of the Hurst exponent is carried out based on the rescaled range method; and its application is extended to a dynamic estimation between the 2006-2021 period. This indicator serves as a market efficiency index and is estimated for the series of daily returns of the MILA securities markets, made up of Chile, Colombia, Mexico, and Peru. The results show that Peru is the least efficient market, and with the highest number of inefficiency cycles for the calculated period. On the other hand, Mexico turns out to be the only MILA market which has been in the efficient region. / Trabajo de investigación