Stochastic processes and applications

我通过开发新的模型和算法来解决社会和经济上重要的问题。我过去的研究对几家公司产生了积极的影响，导致了一个专利奖和另外三个专利申请（都是由合作公司提出的），为我赢得了CBS节目“数字”的咨询职位，并为我赢得了2001年PIMS全国推广奖。未来的计划包括通过冰芯样本研究气候变化，通过最佳检测威胁和瓶颈来改进网络，通过最佳选择广告和分类观众来解决有针对性的广告问题，以及通过跟踪潜艇和污染来保护水域。与这些目标相关的一些数学创新包括发展滤波理论，这些理论可以处理重尾过程，支持已经建立的鲁棒实现，从单纯（部分微分方程定义）轨迹和观察中估计信号，这些信号是根据复杂的点过程、马尔可夫链和随机环境定义的，而不是存在噪声设置的通常扭曲的信号。在每个应用程序中，我都与致力于感兴趣领域的公司或个人合作。对于基本的数学问题，我将利用布朗特教授和我刚刚开发的同胚方法与其他方法来解决基本的概率问题，如存在鞅问题，紧密性，指数紧密性，弱收敛性，大偏差和长时间行为的一大类随机模型。事实上，我们已经用这些同纯方法完成了一项关于紧性的研究，结果是惊人的：与Ethier和Kurtz（1986）或Jakubowski（1986）的标准结果相比，证明更短，而结果更普遍，具有更大的适用性。另一方面，我将进一步发展由Kurtz、Donnelly和Xiong教授介绍的“向下看”粒子表示，并研究其在随机偏微分方程、测量值分支过程、人口模型、受控马尔可夫过程和过滤问题中的应用。尽管Kurtz在1998年发表了论文，但一般粒子表示定理尚未建立。

I solve socially and economically important problems by developing novel models and algorithms. My past research has affected several companies positively, resulted in one patent award with three more filings (all by partner companies), earned me a consulting role with CBS show `Numbers', and won me the 2001 PIMS national outreach award. Future plans include studying climate change through ice core samples, improving the network by detecting threats and bottlenecks optimally, solving targeted advertizing by optimally selecting ads and classifying viewership and securing waters by tracking submarines and pollution. Some mathematical novelties associated with these goals include developing filtering theories that can handle heavy-tailed processes, support robust implementations beyond those already established, estimate signals from mere (partial-differential-equation-defined) traces and observations that are defined in terms of complicated point processes, Markov chains and random environments instead of the usually distorted signal in presence of noise setting. In every application, I work with corporations or individuals dedicated to the area of interest.   For fundamentally mathematical problems, I will utilize homeomorphic methods Prof. Blount and I just developed with other methods to settle basic probabilistic problems such as existence to martingale problems, tightness, exponential tightness, weak convergence, large deviations and long time behaviour for a large class of stochastic models. Indeed, we have already completed a study on tightness using these homeomorphic methods and the results were amazing: the proofs were shorter while the results were far more general, allowing greater applicability, than the standard results in Ethier and Kurtz (1986) or Jakubowski (1986). In another vein, I will further develop the `look down' particle representation, introduced by Professors Kurtz, Donnelly and Xiong, and investigate its application to stochastic partial differential equations, measure-valued branching processes, population models, controlled Markov process and filtering problems. Notwithstanding Kurtz's 1998 paper, the general particle representation theorem has not been established.