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

冲动控制问题在金融市场交易中无处不在。一个突出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拟变分不等式，通常不允许解析解。然而，目前的数值方法，大多不包括自适应技术。自适应离散化是有效和准确地确定最佳交易策略的关键。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处理求解器联合收割机相结合，以获得求解拟变分不等式的有效算法。在跳跃市场中出现的积分项将被包括在内。除此之外，当前金融市场的动荡，如信贷危机和欧洲金融危机，使信贷和交易对手风险增加成为焦点。这就要求对现有模式进行扩展。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之而来的问题作为脉冲控制问题，例如在投资组合优化。他们的解决方案将要求那些有效的和自适应的数值方法的拟变分不等式，这是开发的项目的一部分。最后，在实际应用中，需要利用历史数据对模型参数进行估计。这一额外的困难将通过从不完整的信息中扩展现有的方法来解决。上述办法将在这方面加以推广。这是必要的，以保证开发的计划的实际适用性。

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.