Probability, finance, and SPDE

概率、金融和 SPDE

• 批准号: 8813-2006
• 负责人: Walsh, John
• 金额: $ 1.09万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=339929
• 关键词: Probability; finance; SPDE

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?

超布朗运动是在许多物理情况下出现的重要过程。  它满足奇异的 SPDE，这是现代技术能够处理的边界。  我将研究这个特定方程数值解的某些方面。  我预计它的行为将为解决几个悬而未决的问题提供重要的见解。第二个主题是时间观念反转时 SPDE 的行为。  这将与 Robert Dalang 合作完成。在数学金融学中——它与 SPDE 共享许多概率技术——我将研究期限利率的 Heath-Jarrow-Morton 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，该数字是根据经验选择的。  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项式树方案收敛率的证明扩展到美式导数。  这看起来很容易，但事实并非如此。  有几种可行的方法。  有人对纯概率问题提出了一种有趣的概括：当过滤扩大时，斯涅尔包络线如何变化？