The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first few flips.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19470号 / 理博第4130号 / 新制||理||1594(附属図書館) / 32506 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 熊谷 隆, 教授 岡本 久, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20887号 / 理博第4339号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 福島 竜輝, 教授 中島 啓, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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We focus on planar Random Walks and some related stochastic processes. The discrete models are introduced and some of their core properties examined. We then turn to the question of continuous analogues, starting with the well-known convergence of the Random Walk to Brownian Motion. For the Harmonic Explorer and the Loop Erased Random Walk, we discuss the idea for convergence to SLE(\kappa) and carry out parts of the proof in the former case using a martingale observable to pin down the Loewner driving process.

This thesis examines whether the level of Swedish economic policy uncertainty (EPU) can predict excess returns on the Swedish stock market. We run out-of-sample forecasting using an EPU-based predictive model constructed with the official Swedish EPU index developed by Armelius et al. (2017). Forecasting errors for one-, two-, three-, six-, and twelve-month holding periods and four measures of central tendency are analysed and compared against a random walk benchmark. The findings suggest that EPU has limited forecasting ability for excess stock returns in Sweden, and the EPU-based model demonstrates superior forecasting accuracy only in two out of twenty instances, both for the one-month holding period. However, the forecast errors remain relatively large, casting doubt on the model's ability to outperform the market. Furthermore, the EPU-based model consistently underestimates excess returns, questioning its usefulness as a predictor. Notably, the random walk benchmark's forecast error improves with longer holding periods, raising doubts about the predictability of market movements in the long term.

Search engines play a very important role in daily life. As multimedia product becomes more and more popular, people have developed search engines for images and videos. In the first part of this thesis, I propose a prototype of a book image search engine. I discuss tag representation for the book images, as well as the way to apply the probabilistic model to generate image tags. Then I propose the random walk refinement method using tag similarity graph. The image search system is built on the Galago search engine developed in UMASS CIIR lab. Consider the large amount of data the search engines need to process, I bring in cloud environment for the large-scale distributed computing in the second part of this thesis. I discuss two models, one is the MapReduce model, which is currently one of the most popular technologies in the IT industry, and the other one is the Maiter model. The asynchronous accumulative update mechanism of Maiter model is a great fit for the random walk refinement process, which takes up 84% of the entire run time, and it accelerates the refinement process by 46 times.

A stochastic method is developed, implemented and investigated here for solving Laplace, Poisson's, and standard parabolic wave equations. This method is based on the properties of random walk, diffusion process, Ito formula, Dynkin formula and Monte Carlo simulations. The developed method is a local method i:e: it gives the value of the solution directly at an arbitrary point rather than extracting its value from complete field solution and thus is inherently parallel. Field computation by this method is demonstrated for electrostatic and electrodynamic propagation problems by considering simple examples and numerical results are presented to validate this method. Numerical investigations are carried out to understand efficacy and limitations of this method and to provide qualitative understanding of various parameters involved in this method.

Magister Commercii - MCom / The efficient market hypothesis (EMH) is a controversial theory in Finance. Advocates of the EMH argue that it provides a basis for understanding financial markets while critics suggest that the hypothesis is unreasonable in its assumptions of the real function of these markets. Although the EMH may not be perfect, it provides a sufficient baseline against which financial markets may be analysed. Over the past couple of years, academics have broadly examined the EMH in both developing and developed financial markets. However, limited research has been done on African markets. Therefore, this study examines the weak-form EMH of the Johannesburg Stock Exchange (JSE) after 2008 to ascertain the impact the 2008 global financial crisis had on its efficiency. This study analysed the JSE using weekly and monthly returns of the three major indices (RESI 10, FINI 15, INDI 25) as well as the individual companies under these indices from 30th January 2009 to 30th January 2019. Analysis was carried using various statistical tests i.e., runs test, variance ratio test, unit root tests, and a GARCH model which revealed mixed results. Results of the unit root tests (ADF and PP) confirm that the JSE is weak-form efficient when both the weekly and monthly data of the indices and individual companies are analysed. The results of the runs test reveal that all the weekly and monthly data apart from the weekly data of the companies under RESI 10 index exhibit weak-form efficiency. The variance ratio test confirms weak-form inefficiency when weekly data is used while the monthly data confirms weak form efficiency of the JSE and shows that the market moves from periods of efficiency to periods of relative predictability. The results of the GARCH model on the other hand confirm the weak-form efficiency of the JSE when both the weekly and monthly data of the indices are analysed.

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