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

该提案引入了有助于现代金融数学理论和在概率环境中使用信号处理方法的创新想法、概念和技术。 应用领域包括金融数学中的套利、套期保值和定价，以及对冲基金图像处理和非线性回归中的一些主题。套利是指在没有风险的情况下获利的可能性，广义而言，套期保值意味着重新调整投资组合头寸，以履行未来的财务义务。这些关键的基本概念被用来建立数学市场 模型；尤其是，它们为金融工具提供了衍生的价格概念。我们的一个研究项目描述了一种新的非概率方法来处理套利、对冲和定价的基本概念，该方法强调这样的问题：展开的金融股股票图表的属性是什么？这个问题是在没有考虑路径在某个随机模型中可能具有的概率的情况下处理的。拟议的观点对金融产品价格的可能范围以及风险分析具有经验意义。这些影响中的一些将在拟议的研究中得到发展。 另一项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了新的数学工具。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，并将其提供给研究界。一个相关的项目提议在金融背景下利用新的信号处理方法--压缩传感，以便在观察稀少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 market models; 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.