Stochastic Processes and Applications

该建议包括：1)将过程复制到紧致度量空间进行系统研究；2)研究具有新特征的粒子系统和测值过程；3)设计更好的粒子算法。将通过垂直整合获得协同效应和利益。动机来自跟踪、预测、信号处理、疾病演变、欺诈检测、广告、信息传播、天气预报、金融、保险、随机方程模拟以及长记忆、随机环境和不良空间的数学。1)非线性滤子、鞅问题、随机偏微分方程或Dirichlet形式的存在唯一性；2)用这些方法描述的平稳过程的存在唯一性；3)有限维分布、渐进过程、CadLag过程或近似滤子的紧性和收敛；或4)马尔可夫过程、Dirichlet形式、鞅问题或大偏差的一般理论；大多仅适用于Polish或紧空间。一个人在个案的基础上处理奇异空间的问题。我们将一般存在、唯一性、紧性、收敛、平稳分布、马尔可夫过程、Dirichlet形式、过滤和大偏差结果从紧的Polish空间推广到相当一般的拓扑空间。超过程、Fleming-Viot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、分支和竞争的个体在生物学上驱动的，但大多缺乏真实世界的特征。例如，观测个体基本上消失，然后在其他地方重新出现，已经用(随机方向和)Levy飞行时间建模，但在首选位置有山谷的恶劣环境可能比只有厚重的尾巴更好地解释观测。我们将研究新的和现有的粒子和测量值过程的适定性、支持性、长时间行为和便于模拟的表示。序列蒙特卡罗(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，使用缓慢且不准确的欧拉或米尔斯坦方法。或者，可以不正确但快速地模拟重要采样权重以更正，但会出现权重差异和有效退化到较少粒子。SMC是一种解决方案：我们交互、分支或重采样到均匀的权重，保留路径空间定价的原始路径。我们将产生新的SMC算法、应用程序和分析。

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.