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

在金融市场交易时，冲动控制问题无处不在。一个明显的例子是有交易成本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，这通常不允许有解析解。然而，目前的数值方法大多不包含自适应技术。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，并将它们与有效的预条件求解器相结合，以获得求解拟变分不等式的有效算法。积分项，出现在跳跃的市场中，将被包括在内。此外，当前金融市场的动荡，如信贷危机和欧洲金融危机，将重点放在增加信贷和交易对手风险上。这就要求延长现有的模型。马尔可夫因素模型是实现金融市场复杂动态的低维表示的合适工具。我们将把随之而来的问题表述和分析为冲动控制问题，例如在投资组合优化方面。他们的解决方案将需要那些有效的和自适应的拟变分不等式的数值方法，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是，实际应用需要利用历史数据来估计模型参数。这个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。上述办法将向这一方向延伸。这对于保证所开发方案的实际适用性是必要的。

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.