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

函数Itô-calculus是在过去8年中发展起来的，它是对Itô-formula的深远推广，而Itô-formula是随机分析的基础。在数学上，它提供了半鞅函数的半鞅分解的显式形式。在项目中，该函数Itô-calculus应扩展到超过程，这是一类重要的无限维测度值随机过程。最初，他们的动机是生物学应用，但同时，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，基于超过程的函数Itô-calculus，我们想在矩阵表示中推导出被积函数的直接表示。在项目应用较多的部分，我们分析了一种特定类型的信用风险，即交易对手风险。如果衍生合约中的交易对手没有履行其付款义务，就会发生违约。在超过程的基础上，我们想为路径依赖的衍生产品开发一个定价公式，该公式考虑了交易对手风险。

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.