Probability, finance, and SPDE

超布朗运动是一个重要的过程，在许多物理情况下出现。 它满足一个奇异的SPDE，这是在现代技术将处理的边界上。 我将研究这个特殊方程的数值解的某些方面。 我希望它的行为将为几个悬而未决的问题提供重要的见解。第二个主题是当时间感颠倒时SPDE的行为。 这将与罗伯特·达朗合作完成。在数学金融学中-它与SPDE共享许多概率技术-我将研究Heath-Jarrow-Morton长期利率模型的复杂性。 这里的“复杂性”是指它所包含的独立因素的数量。 这在模型中没有确定，在实际情况中，这个数字是根据经验选择的。 我建议制定一些更确切的准则。最后一个问题是将我的二叉树格式的收敛速度的证明推广到美国型导数。 这看起来很容易，但事实并非如此。 有几种可行的办法。 其中一个建议一个有趣的概括问题的纯概率：如何斯内尔信封改变时，过滤扩大？

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?