Probability, finance, and SPDE

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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?