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

超级过程的函数 IT 演算以及超级过程在交易对手风险中的应用

• 批准号: 388370633
• 负责人: Professor Dr. Ludger Overbeck
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/388370633?language=en
• 关键词: Functional; calculus; superprocesses; application; counterparty

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.

函数伊藤演算，在过去的8年中发展起来，是伊藤公式的一个深远的推广，这是随机分析的基础。在数学上，它提供了半鞅的函数的半鞅分解的显式形式。在该项目中，这一功能伊藤演算应扩展到超过程，一类重要的无限维测度值随机过程。最初，他们的动机是生物应用，但同时他们发现自己的方式进入数学金融。该项目的一个目标是获得超过程的鞅表示，这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-演算，我们希望得到被积函数在鞅表示中的直接表示。在项目的应用部分，我们分析了一种特定类型的信用风险，即交易对手风险。如果衍生工具合约中的交易对手没有履行其付款义务，就会发生这种情况。基于超过程，我们希望开发一个定价公式的路径依赖的衍生产品，其中考虑到对手方的风险。