This paper explores stochastic modeling approaches to elucidate the intricate dynamics of stock prices and volatility in financial markets. Beginning with an overview of Brownian motion and its historical significance in finance, we delve into various stochastic models, including the classic Black-Scholes framework, the Heston model, fractional Brownian motion, GARCH models, and Levy processes. Through a thorough investigation, we analyze the strengths and limitations of each model in capturing key features of financial time series data. Our empirical analysis focuses on parameter estimation and model calibration using Levy processes, demonstrating their effectiveness in predicting stock returns. However, we acknowledge the need for further refinement and exploration, suggesting potential avenues for future research, such as hybrid modeling approaches. Overall, this study underscores the importance of stochastic modeling in understanding market dynamics and informs decision-making in the financial industry.
本文探讨了随机建模方法，以阐明金融市场中股票价格和波动的复杂动态。本文从布朗运动及其在金融学中的历史意义开始，深入探讨了各种随机模型，包括经典的布莱克-斯科尔斯框架、海斯顿模型、分数布朗运动、GARCH 模型和列维过程。通过深入研究，我们分析了每种模型在捕捉金融时间序列数据关键特征方面的优势和局限。我们的实证分析侧重于使用 Levy 过程进行参数估计和模型校准，证明了它们在预测股票收益方面的有效性。不过，我们也承认需要进一步完善和探索，并提出了未来研究的潜在途径，如混合建模方法。总之，本研究强调了随机建模在理解市场动态和为金融业决策提供信息方面的重要性。
{"title":"A Basic Overview of Various Stochastic Approaches to Financial Modeling With Examples","authors":"Aashrit Cunchala","doi":"arxiv-2405.01397","DOIUrl":"https://doi.org/arxiv-2405.01397","url":null,"abstract":"This paper explores stochastic modeling approaches to elucidate the intricate\u0000dynamics of stock prices and volatility in financial markets. Beginning with an\u0000overview of Brownian motion and its historical significance in finance, we\u0000delve into various stochastic models, including the classic Black-Scholes\u0000framework, the Heston model, fractional Brownian motion, GARCH models, and Levy\u0000processes. Through a thorough investigation, we analyze the strengths and\u0000limitations of each model in capturing key features of financial time series\u0000data. Our empirical analysis focuses on parameter estimation and model\u0000calibration using Levy processes, demonstrating their effectiveness in\u0000predicting stock returns. However, we acknowledge the need for further\u0000refinement and exploration, suggesting potential avenues for future research,\u0000such as hybrid modeling approaches. Overall, this study underscores the\u0000importance of stochastic modeling in understanding market dynamics and informs\u0000decision-making in the financial industry.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jingsi Hou, Guangyan Huang, Sammy Suliman, Haoran Yan
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as combinatorics, graph theory, and theoretical computer science. In this paper, we seek to provide an introduction to the relevant topological concepts to non-specialists, as well as a selection of some existing applications to theorems in discrete mathematics.
{"title":"Lovasz' Conjecture and Other Applications of Topological Methods in Discrete Mathematics","authors":"Jingsi Hou, Guangyan Huang, Sammy Suliman, Haoran Yan","doi":"arxiv-2405.05273","DOIUrl":"https://doi.org/arxiv-2405.05273","url":null,"abstract":"In 20th century mathematics, the field of topology, which concerns the\u0000properties of geometric objects under continuous transformation, has proved\u0000surprisingly useful in application to the study of discrete mathematics, such\u0000as combinatorics, graph theory, and theoretical computer science. In this\u0000paper, we seek to provide an introduction to the relevant topological concepts\u0000to non-specialists, as well as a selection of some existing applications to\u0000theorems in discrete mathematics.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the problems and solutions to the 50th Annual USA Mathematical Olympiad.
我们介绍了第 50 届美国数学奥林匹克竞赛的问题和解决方案。
{"title":"Report on the 50th Annual USA Mathematical Olympiad","authors":"Bela Bajnok","doi":"arxiv-2406.09518","DOIUrl":"https://doi.org/arxiv-2406.09518","url":null,"abstract":"We present the problems and solutions to the 50th Annual USA Mathematical\u0000Olympiad.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the problems and solutions to the 12th Annual USA Junior Mathematical Olympiad.
我们介绍第 12 届美国青少年数学奥林匹克竞赛的问题和解决方案。
{"title":"Report on the 12th Annual USA Junior Mathematical Olympiad","authors":"Bela Bajnok, Evan Chen","doi":"arxiv-2406.11094","DOIUrl":"https://doi.org/arxiv-2406.11094","url":null,"abstract":"We present the problems and solutions to the 12th Annual USA Junior\u0000Mathematical Olympiad.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the problems and solutions to the 61st Annual International Mathematical Olympiad
我们介绍第 61 届国际数学奥林匹克年会的问题和解决方案
{"title":"Report on the 61st Annual International Mathematical Olympiad","authors":"Bela Bajnok, Evan Chen","doi":"arxiv-2406.09517","DOIUrl":"https://doi.org/arxiv-2406.09517","url":null,"abstract":"We present the problems and solutions to the 61st Annual International\u0000Mathematical Olympiad","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning. And this is also typical for mathematics: the solution to a problem studied out of pure curiosity turns out to be useful in representing real objects and processes that are completely different in nature. Despite the published machine methods for combinatorial proof of the Four color conjecture, there is still no clear description of the mechanism for coloring a planar graph with four colors, its natural essence and its connection with the phenomenon of graph planarity. It is necessary not only to prove (preferably by deductive methods) that any planar graph can be colored with four colors, but also to show how to color it. The paper considers an approach based on the possibility of reducing a maximally flat graph to a regular flat cubic graph with its further coloring. Based on the Tate-Volynsky theorem, the vertices of a maximally flat graph can be colored with four colors, if the edges of its dual cubic graph can be colored with three colors. Considering the properties of a colored cubic graph, it can be shown that the addition of colors obeys the transformation laws of the fourth order Klein group. Using this property, it is possible to create algorithms for coloring planar graphs.
{"title":"Algorithmic methods of finite discrete structures. The Four Color Theorem. Theory, methods, algorithms","authors":"Sergey Kurapov, Maxim Davidovsky","doi":"arxiv-2405.05270","DOIUrl":"https://doi.org/arxiv-2405.05270","url":null,"abstract":"The Four color problem is closely related to other branches of mathematics\u0000and practical applications. More than 20 of its reformulations are known, which\u0000connect this problem with problems of algebra, statistical mechanics and\u0000planning. And this is also typical for mathematics: the solution to a problem\u0000studied out of pure curiosity turns out to be useful in representing real\u0000objects and processes that are completely different in nature. Despite the\u0000published machine methods for combinatorial proof of the Four color conjecture,\u0000there is still no clear description of the mechanism for coloring a planar\u0000graph with four colors, its natural essence and its connection with the\u0000phenomenon of graph planarity. It is necessary not only to prove (preferably by\u0000deductive methods) that any planar graph can be colored with four colors, but\u0000also to show how to color it. The paper considers an approach based on the\u0000possibility of reducing a maximally flat graph to a regular flat cubic graph\u0000with its further coloring. Based on the Tate-Volynsky theorem, the vertices of\u0000a maximally flat graph can be colored with four colors, if the edges of its\u0000dual cubic graph can be colored with three colors. Considering the properties\u0000of a colored cubic graph, it can be shown that the addition of colors obeys the\u0000transformation laws of the fourth order Klein group. Using this property, it is\u0000possible to create algorithms for coloring planar graphs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"186 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper aims to explore the evolution of image denoising in a pedagological way. We briefly review classical methods such as Fourier analysis and wavelet bases, highlighting the challenges they faced until the emergence of neural networks, notably the U-Net, in the 2010s. The remarkable performance of these networks has been demonstrated in studies such as Kadkhodaie et al. (2024). They exhibit adaptability to various image types, including those with fixed regularity, facial images, and bedroom scenes, achieving optimal results and biased towards geometry-adaptive harmonic basis. The introduction of score diffusion has played a crucial role in image generation. In this context, denoising becomes essential as it facilitates the estimation of probability density scores. We discuss the prerequisites for genuine learning of probability densities, offering insights that extend from mathematical research to the implications of universal structures.
{"title":"Denoising: from classical methods to deep CNNs","authors":"Jean-Eric Campagne","doi":"arxiv-2404.16617","DOIUrl":"https://doi.org/arxiv-2404.16617","url":null,"abstract":"This paper aims to explore the evolution of image denoising in a\u0000pedagological way. We briefly review classical methods such as Fourier analysis\u0000and wavelet bases, highlighting the challenges they faced until the emergence\u0000of neural networks, notably the U-Net, in the 2010s. The remarkable performance\u0000of these networks has been demonstrated in studies such as Kadkhodaie et al.\u0000(2024). They exhibit adaptability to various image types, including those with\u0000fixed regularity, facial images, and bedroom scenes, achieving optimal results\u0000and biased towards geometry-adaptive harmonic basis. The introduction of score\u0000diffusion has played a crucial role in image generation. In this context,\u0000denoising becomes essential as it facilitates the estimation of probability\u0000density scores. We discuss the prerequisites for genuine learning of\u0000probability densities, offering insights that extend from mathematical research\u0000to the implications of universal structures.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper aims to give a brief account of the mathematical work of the 7th-century Armenian polymath and natural philosopher Anania Shirakatsi. The three sections of Anania's ``Book of Arithmetic'' -- tables of arithmetic operations, a list of problems and their answers, and a collection of entertaining puzzles -- are presented and discussed, the focus being on the problems and solutions. A close examination of the structure of these problems reveals that Anania's proficiency in arithmetic was considerably more sophisticated than their mathematical contents might suggest. The geography of Anania's problems is illustrated through two maps highlighting the locations referenced within these problems.
{"title":"The mathematical work of Anania Shirakatsi","authors":"Vahagn Aslanyan","doi":"arxiv-2404.15945","DOIUrl":"https://doi.org/arxiv-2404.15945","url":null,"abstract":"This paper aims to give a brief account of the mathematical work of the\u00007th-century Armenian polymath and natural philosopher Anania Shirakatsi. The\u0000three sections of Anania's ``Book of Arithmetic'' -- tables of arithmetic\u0000operations, a list of problems and their answers, and a collection of\u0000entertaining puzzles -- are presented and discussed, the focus being on the\u0000problems and solutions. A close examination of the structure of these problems\u0000reveals that Anania's proficiency in arithmetic was considerably more\u0000sophisticated than their mathematical contents might suggest. The geography of\u0000Anania's problems is illustrated through two maps highlighting the locations\u0000referenced within these problems.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A brief overview and history of the American Mathematics Competitions.
美国数学竞赛简介和历史。
{"title":"The AMC -- What It Is and Why It Matters","authors":"Bela Bajnok","doi":"arxiv-2404.15511","DOIUrl":"https://doi.org/arxiv-2404.15511","url":null,"abstract":"A brief overview and history of the American Mathematics Competitions.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This biographical and scientific memoir of Dominic Welsh includes summaries of his important contributions to probability and combinatorics. With John Hammersley, he introduced first-passage percolation, and in so doing they formulated and proved the first subadditive ergodic theorem. Welsh has numerous results in matroid theory, and wrote the first monograph on the topic. He worked on computational complexity and particularly the complexity of computing the Tutte polynomial. He was an inspirational teacher and advisor who helped to develop a community of scholars in combinatorics.
{"title":"Dominic Welsh (1938-2023)","authors":"Geoffrey R. Grimmett","doi":"arxiv-2404.13942","DOIUrl":"https://doi.org/arxiv-2404.13942","url":null,"abstract":"This biographical and scientific memoir of Dominic Welsh includes summaries\u0000of his important contributions to probability and combinatorics. With John\u0000Hammersley, he introduced first-passage percolation, and in so doing they\u0000formulated and proved the first subadditive ergodic theorem. Welsh has numerous\u0000results in matroid theory, and wrote the first monograph on the topic. He\u0000worked on computational complexity and particularly the complexity of computing\u0000the Tutte polynomial. He was an inspirational teacher and advisor who helped to\u0000develop a community of scholars in combinatorics.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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