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

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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.