Impulse Control Problems and Adaptive Numerical Solution of Quasi-Variational Inequalities in Markovian Factor Models

马尔可夫因子模型中拟变分不等式的脉冲控制问题和自适应数值解

• 批准号: 265374484
• 负责人: Professor Dr. Roland Herzog
• 金额: --
• 依托单位: Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/265374484?language=en
• 关键词: Impulse; Control; Problems; Adaptive; Numerical

Impulse control problems are ubiquitous when trading in financial markets. One striking example are portfolio optimization problems under transaction costs. Their mathematical description leads to quasi-variational inequalities, which typically do not admit analytical solutions. Current numerical methods, however, mostly do not incorporate adaptive techniques. Adaptive discretization is the key to an efficient and accurate determination of optimal trading strategies. In this project we shall develop adaptive methods and combine them with effective preconditioned solvers to obtain an efficient algorithm for the solution of quasi-variational inequalities. Integral terms, which occur in markets with jumps, will be included. Besides this, the current turbulence in financial markets, such as the credit crisis and the European financial crisis, puts the focus on increasing credit and counterparty risk. This asks for an extension of existing models. Markovian factor models are an appropriate tool to achieve a low-dimensional representation of complex dynamics on financial markets. We will formulate and analyze the ensuing problems as impulse control problems, for instance in portfolio optimization. Their solution will ask for those efficient and adaptive numerical methods for quasi-variational inequalities, which are developed as part of the project. Last but not least, the practical application requires the estimation of model parameters by means of historical data. This additional difficulty will be tackled by extending existing methodologies from incomplete information. The above-mentioned approaches will be extended in this direction. This is necessary to guarantee the practical applicability of the developed schemes.

在金融市场交易时，冲动控制问题普遍存在。一个引人注目的例子是交易成本下的投资组合优化问题。它们的数学描述导致了准变分不等式，通常不允许解析解。然而，当前的数值方法大多不结合自适应技术。自适应离散化是高效、准确地确定最优交易策略的关键。在这个项目中，我们将开发自适应方法并将其与有效的预处理求解器相结合，以获得求解拟变分不等式的有效算法。将包括出现在跳跃市场中的积分项。此外，当前信用危机、欧洲金融危机等金融市场动荡，使得信用风险和交易对手风险加剧。这需要对现有模型进行扩展。马尔可夫因子模型是实现金融市场复杂动态的低维表示的合适工具。我们将制定和分析随后的问题作为脉冲控制问题，例如在投资组合优化中。他们的解决方案将需要那些高效且自适应的准变分不等式数值方法，这些方法是作为该项目的一部分开发的。最后但并非最不重要的一点是，实际应用需要借助历史数据来估计模型参数。这一额外的困难将通过从不完整的信息扩展现有的方法来解决。上述方法将朝这个方向延伸。这是保证所开发方案的实际适用性所必需的。