"Non Probabilistic Financial Mathematics. Discretization of Processes, Wavelets and Applications."

该提案介绍了有助于现代金融数学理论和在概率设置中使用信号处理方法的创新思想、概念和技术。应用领域是金融数学中的套利、套期保值和定价，以及对冲基金的图像处理和非线性回归等一些主题。套利是指在没有风险的情况下获利的可能性，套期保值从广义上讲是指重新调整投资组合的头寸，以满足未来的财务义务。这些关键的基本概念用于建立数学市场模型；特别是，它们为金融工具提供了派生的价格概念。我们的一个研究项目描述了一种新的非概率方法来研究套利、对冲和定价的基本概念，该方法强调的问题类型是：金融股的展开股票图表的属性是什么？在处理这个问题时，不考虑路径在某个随机模型中可能具有的概率。提出的观点对金融产品价格的可能范围以及风险分析具有实证意义。其中一些影响将在拟议的研究中得到发展。另一个研究项目为图像和视频处理以及二维和三维图像中的细胞识别开发新的数学工具。这个项目导致了技术创新的进步，主要目标是在软件中实现这些新的算法，这些算法将提供给研究社区。一个相关的项目建议在金融环境中使用压缩感知的新信号处理方法，以便在缺乏观测值的情况下估计投资组合的行为。该方法提供了几个潜在的优势，因为它考虑了风险因素的联合概率。

The proposal introduces innovative ideas, concepts and techniques that contribute to the theory of modern financial mathematics and the use of signal processing methods in probabilistic settings.The application areas are arbitrage, hedging and pricing in financial mathematics as well as some topics from image processing and nonlinear regression of hedge funds. Arbitrage refers to the possibility of making a profit without risk and hedging, in broad terms, means the re-adjustment of portfolio positions to meet future financial obligations. These key foundational concepts are used to build mathematical marketmodels; in particular, they provide the derived notion of price for financial instruments. One of our research projects describes a new non probabilistic approach to the foundational notions of arbitrage, hedging and pricing, the approach emphasizes questions of the type: what are the properties of an unfolding stock chart for a financial stock? This question is dealt with without considering the probability that the path may have in a certain stochastic model. The proposed point of view carries empirical implications for the possible range of prices of financial products as well as risk analysis. Some of these implications will be developed in the proposed research.Another research project develops new mathematical tools for image and video processing as well as cell recognition in two and three dimensional images. This project leads to advances in technological innovations, a main goal is to implement these new algorithms in software that will be made available to the research community. A related project proposes to make use of the new signal processing methodology of compressed sensing in a financial context in order to estimate the behavior of a portfolio in the presence of scarce observations. The methodology offers several potential advantages as it accounts for the joint probability of risk factors.