Functional Itô-calculus for superprocesses and application of superprocesses to counterparty risk

近8年发展起来的泛函itó-演算是随机分析的基础--itó-公式的一个深远的推广。在数学上，它给出了半鞅的函数的半鞅分解的一种显式形式。在这个项目中，这个泛函演算将被推广到超过程，这是一类重要的无限维测值随机过程。最初，他们的动机是生物应用，但与此同时，他们找到了进入数学金融的途径。该项目的一个目标是获得超过程的一个鞅表示，这通常是随机分析中的另一个基本结果。特别地，基于超过程的泛函演算，我们想要得到被积函数在被积函数表示下的直接表示。在项目的应用较多的部分，我们分析了一种特定类型的信用风险，即交易对手风险。如果衍生品合同的对手方没有完全履行其付款义务，就会发生这种情况。基于超过程，我们想要建立一个考虑交易对手风险的路径依赖衍生产品的定价公式。

Functional Itô-calculus, developed in the last 8 years, is a far reaching generalization of the Itô-formula, which is fundamental for Stochastic Analysis. Mathematically, it provides an explicit form of the semimartingal decomposition of a function of a semimartingale. In the project this functional Itô-calculus shall be extended to superprocesses, an important class of infinite-dimensional measure-valued stochastic processes. Originally they were motivated by biological applications, but meanwhile they found their way into mathematical finance. One objective of the project is to obtain a martingal representation for superprocesses, which is in general another fundamental results in Stochastic Analysis. In particular, based on the functional Itô-calculus for superprocesses we want to derive a direct representation of the integrand in the martingal representation. In the more applied part of the project we analyse a specific type of credit risk, namly the counterparty risk. It occurs if a counterparty in a derivative contract does not fullfill his or her payment obligations. Based on superprocesses we want to develop a pricing formula for path-dependent derivative products, which takes into account the counterparty risk.