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

• Applicant: Professor Dr. Ludger Overbeck
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 388370633

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.

• DFG Programme: Research Grants