关于金融数学中的三个过滤问题
On three filtering problems arising in mathematical finance
摘要：
在三种情况下，过滤理论用于金融数学在不同层次的细节说明。这三个问题源于以下不同的工作：1）关于从双边汇率新闻中估计随机波动率模型，R。Mahieu，and P. Schotman; 2）状态空间方法估计利率期限结构的多因素CIR模型，由A. L. J. Geyer和S. Pichler; 3）金融衍生品定价中部分观测下的风险最小化套期保值策略。Platen和W. J. Runggaldier;在第一个问题中，我们建议使用最近的非线性滤波技术的基础上几何估计波动时间序列从观察到的双边汇率。这里使用的模型是随机波动率模型。我们提出的滤波器被称为投影滤波器，并给出了这种滤波器的简要推导。详细介绍了第二个问题，并暗示了可能使用不同的过滤技术。事实上，在2）和部分文献中用于这个问题的滤波器可以被解释为投影滤波器，我们将对如何构造更一般和可能更合适的投影滤波器做一些评论。第三个问题只是简短地提出。
英文摘要：
Three situations in which filtering theory is used in mathematical finance are illustrated at different levels of detail. The three problems originate from the following different works: 1) On estimating the stochastic volatility model from observed bilateral exchange rate news, by R. Mahieu, and P. Schotman; 2) A state space approach to estimate multi-factors CIR models of the term structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3) Risk-minimizing hedging strategies under partial observation in pricing financial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In the first problem we propose to use a recent nonlinear filtering technique based on geometry to estimate the volatility time series from observed bilateral exchange rates. The model used here is the stochastic volatility model. The filters that we propose are known as projection filters, and a brief derivation of such filters is given. The second problem is introduced in detail, and a possible use of different filtering techniques is hinted at. In fact the filters used for this problem in 2) and part of the literature can be interpreted as projection filters and we will make some remarks on how more general and possibly more suitable projection filters can be constructed. The third problem is only presented shortly.
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