Probability, finance, and SPDE

超布朗运动是在许多物理情况下出现的一个重要过程。如果它满足一个奇异的SPDE，这是现代技术可以处理的边缘。我将研究这个特殊方程的数值解的某些方面。我希望它的行为将给几个公开的问题带来重要的洞察。第二个话题是当时间感颠倒时SPDEs的行为。这将与Robert Dalang合作完成。在数学金融学中-它与SPDEs分享了许多概率技术-我将研究Heath-Jrow-Morton定期利率模型的复杂性。这里的“复杂性”是指它包含的独立因素的数量。*这不是在模型中确定的，在实际情况中，数字是根据经验法则选择的。我建议制定一些更准确的指导方针。最后一个问题是将我对二叉树格式收敛速度的证明推广到美式导数。这看起来很容易，但实际上并非如此。有几种可行的方法。其中一种方法对一个纯粹概率的问题提出了一个有趣的推广：当过滤扩大时，Snell包络是如何变化的？

Super Brownian motion is an important process which arises in numerous physical situations.  It satisfies a singular SPDE, which is on the borderline of what modern techniques will handle.  I will study some aspects of the numerical solution of this particular equation.  I expect that its behavior will give important insight into several open problems. A second topic is the behavior of SPDEs when the sense of time is reversed.  This will be done in collaboration with Robert Dalang. In mathematical finance---which shares a number of probabilistic techniques with SPDEs---I will study the complexity of the Heath-Jarrow-Morton model of term interest rates.  "Complexity" here is in terms of the number of independent factors it contains.  This is not determined in the model, and in practical situations, the number is chosen by rule of thumb.  I propose to develop some more exact guidelines. The last problem is to extend my proof of the rate of convergence of the binomial tree scheme to American type derivatives.  This appears easy, but is not.  There are several feasible approaches.  One suggests an intriguing generalization to a problem in pure probability: how does a Snell envelope change when the filtration is enlarged?