Stochastic Processes and Applications

随机过程及其应用

• 批准号: RGPIN-2018-05114
• 负责人: Kouritzin, Michael
• 金额: $ 1.46万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=655936
• 关键词: Stochastic; Processes; Applications

The proposal involves: 1) replicating processes onto compact metric spaces for systematic study; 2) studying particle systems and measure-valued processes with new features; and 3) engineering of better particle algorithms. Synergies and interest will be gained through vertical integration. Motivation derives from tracking, prediction, signal processing, disease evolution, fraud detection, spot advertising, information spread, weather prediction, finance, insurance, stochastic equation simulation as well as the mathematics of long memory, random environments and bad spaces. ******General results for the: 1) existence and uniqueness of nonlinear filters, martingale problems, stochastic partial differential equations or Dirichlet forms; 2) existence and uniqueness of stationary processes described these ways; 3) tightness and convergence of finite-dimensional distributions, progressive processes, cadlag processes or approximate filters; or 4) general theory of Markov processes, Dirichlet forms, martingale problems or large deviations; are mostly only for Polish or compact spaces. One treats problems on exotic spaces on a case-by-case basis. We will extend general existence, uniqueness, tightness, convergence, stationary distribution, Markov process, Dirichlet form, filtering and large deviation results from compact Polish spaces to quite general topological spaces. ******Superprocesses, Fleming-Viot processes, interacting particle systems and branching processes are motivated biologically by interacting, branching and competing individuals of possibly multiple types but mostly lack real-world features. For example, the observation individuals basically disappear and then re-appearing elsewhere has been modeled by (random direction and) Levy flight times but a rough environment with valleys at preferred locations may explain the observation better than just heavy tails. We will study well-posedness, support properties, long-time behavior and simulation-facilitating representations for new and existing particle and measure-valued processes.******Sequential Monte Carlo (SMC) methods process big data in real time; are used for classifying, locating, tracking, predicting, model selection and parameter estimation; and have potential new application in medical imaging, fraud detection, hidden liquidity, network security, option pricing, insurance analysis and stochastic convergence. For example, option pricing often uses Monte Carlo simulation with slow and inaccurate Euler or Milstein methods. Alternatively, one can simulate incorrectly but quickly with importance sampling weights to correct but weight discrepancy and effective degeneration to few particles occurs. SMC is a solution: We interact, branch or resample to even weights, keeping the ancestral paths for pathspace pricing. We will produce new SMC algorithms, applications and analysis.

该提案涉及：1）将过程复制到紧凑的度量空间上以进行系统研究； 2）研究具有新特征的粒子系统和测值过程； 3）更好的粒子算法的工程。通过垂直整合将获得协同效应和利益。动机来源于跟踪、预测、信号处理、疾病演化、欺诈检测、广告投放、信息传播、天气预报、金融、保险、随机方程模拟以及长记忆、随机环境和恶劣空间的数学。 ******一般结果：1）非线性滤波器、鞅问题、随机偏微分方程或狄利克雷形式的存在性和唯一性； 2）以这些方式描述的平稳过程的存在性和唯一性； 3) 有限维分布、渐进过程、卡滞过程或近似滤波器的紧密性和收敛性；或4）马尔可夫过程、狄利克雷形式、鞅问题或大偏差的一般理论；大多数情况下仅适用于波兰或紧凑的空间。人们会根据具体情况来处理异域空间中的问题。 我们将把一般存在性、唯一性、紧密性、收敛性、平稳分布、马尔可夫过程、狄利克雷形式、过滤和大偏差结果从紧波兰空间扩展到相当一般的拓扑空间。 ******超级过程、弗莱明-维奥特过程、相互作用的粒子系统和分支过程是由可能多种类型的相互作用、分支和竞争个体在生物学上激发的，但大多缺乏现实世界的特征。例如，观察个体基本上消失，然后在其他地方重新出现，这是通过（随机方向和）利维飞行时间建模的，但在首选位置有山谷的恶劣环境可能比仅仅重尾更好地解释观察结果。 我们将研究新的和现有的粒子和测量值过程的适定性、支持属性、长期行为和促进模拟的表示。******顺序蒙特卡罗（SMC）方法实时处理大数据；用于分类、定位、跟踪、预测、模型选择和参数估计；并在医学成像、欺诈检测、隐藏流动性、网络安全、期权定价、保险分析和随机收敛等方面具有潜在的新应用。例如，期权定价通常使用蒙特卡罗模拟以及缓慢且不准确的欧拉或米尔斯坦方法。另一种方法是，可以错误地但快速地进行模拟，并使用重要性采样权重来纠正，但会发生权重差异和对少数粒子的有效退化。 SMC 是一种解决方案：我们进行交互、分支或重新采样以达到均匀权重，从而保留路径空间定价的祖先路径。我们将产生新的 SMC 算法、应用程序和分析。