Stochastic modelling and network analysis/Modélisation stochastique et analyse de réseaux

拟议的研究由以下两个统一主题下的几个正在进行的项目组成。*主要主题是在非线性和随机性影响整体中大量元素的行为的情况下进行数学建模，例如金融市场上商品价值的波动。在离散案例中，我特别感兴趣的策略是，通过使用导致帕隆多悖论现象的技术，将客户失去的一系列机会逆转为成功的投资组合。在连续情形下，需要通过相互作用的粒子系统建立、模拟和分析伪装成适定鞅问题的非线性随机偏微分方程组(或相关的离散逼近格式)。我在未来几年的目标是继续我正在进行的关于这些系统的基本性质的研究项目，这些项目具有潜在的意义，因为它们将经典极限理论的适用性扩展到新的数学背景。*补充主题是在某些大数据库内的知识发现和数据挖掘的背景下的统计建模，特别是多模式数据的异质信息网络，如社交媒体网络。这些要求确定最适合用于链接预测和社区检测目的的多重比较框架的那些数学特征。我计划将层级结构纳入专门针对这些数据库的评估方法中。这里的新颖性来自于使用先进的数据挖掘和分析技术，包括正式的概念分析，对这些问题提出了新的解决方案。

The proposed research is composed of several ongoing projects, under the following two unifying themes. ******The main theme is mathematical modelling in situations where nonlinearity and randomness affect the behavior of large numbers of elements in an ensemble, such as the fluctuations in value of commodities on the financial markets. In the discrete case I am specifically interested in strategies leading to the reversal from a collection of losing opportunities for the client, into a winning portfolio through the use of techniques giving rise to the phenomenon called Parrondo's paradox. In the continuous case, nonlinear stochastic partial differential equations (or some associated discrete approximation scheme) in the guise of well-posed martingale problems need to be built, simulated and analyzed via interacting particle systems. My objective for the years to come is the pursuit of my ongoing research projects on the fundamental properties of these systems, which bear potential significance as they extend the applicability of classical limit theory to new mathematical contexts. ******The complementary theme is statistical modelling in the context of knowledge discovery and data mining within certain big data bases, notably heterogeneous information networks of multi-modal data such as social media networks. These require the identification of those mathematical characteristics that best suit a framework of multiple comparisons for the purpose of link prediction and community detection. I plan to incorporate hierarchical structures in the estimation methodology specifically destined for these databases. The novelty here comes from the proposal of new solutions to these issues using advanced data mining and analysis techniques, including formal concept analysis.***********