Financial modelling in an HMM-modulated regime-switching framework

这一研究方案围绕金融建模中的体制转换框架这一主题展开。政权的转换由代表“世界状态”的连续或离散时间隐马尔可夫链控制。研究了隐马尔可夫模型(HMM)的新的实际应用和理论发展。首先，将隐马尔可夫滤波理论扩展到处理时间序列数据的长程相关性，说明了隐马尔可夫模型中的马尔可夫假设并不像它最初显现的那样具有限制性。这导致了驱动模型参数的随机演化的弱隐马尔可夫模型，从而捕捉到了制度转换和记忆特性。其次，将卡尔曼滤波等非线性滤波技术与最优HMM滤波理论相结合，在非线性时间序列中的金融变量和其他现象的参数估计和建模中产生了自适应的高性能算法。第三，设计了年金挂钩合同和死亡率挂钩证券的有效估值方法，其中假定金融和人口因素的动态是相关的，利率和死亡率都是HMM调制的。第四，研究了基于隐马尔可夫模型的期权定价和债券定价中时变参数的恢复问题。金融学中的这个逆问题可以通过使用ansatz方法或通过将其重新表述为数学分析中的Stieltjes逆矩问题的推广来解决。在某些情况下，例如在债券定价中，将构造生成函数，它可以被视为债券价格解的一个组成部分的变换。虽然所有这些研究都与定量金融学和精算学有直接关系，但其他应用领域，包括经济学、工程学和其他应用科学分支，也可以从这项研究计划的结果中受益。

This research programme centres on the theme of regime-switching frameworks in financial modelling. The switching of regimes is governed by either a continuous- or discrete-time hidden Markov chain representing the "state of the world". New practical applications and further theoretical developments of hidden Markov models (HMM) are investigated. First, HMM filtering theory is extended to deal with the long range dependence of time series data illustrating that the Markovian assumption in HMM is not as restrictive as it first appears. This gives rise to weak HMMs driving the stochastic evolution of model parameters thereby capturing both regime-switching and memory properties. Second, Kalman filtering and other non-linear filtering techniques are coupled with optimal HMM filtering theory to produce adaptive and high performance algorithms in the parameter estimation and modelling of financial variables and other phenomena in nonlinear time series. Third, methodologies are devised for efficient valuation of annuity-linked contracts and mortality-linked securities where the dynamics of both financial and demographic factors are assumed correlated, and both interest and mortality rates are HMM-modulated. Fourth, the recovery of time-varying parameters for HMM-based models in option pricing and bond valuation is considered. This inverse problem in finance may be addressed either by using an ansatz approach or by reformulating it as a generalisation of the inverse Stieltjes moment problem of mathematical analysis. In some cases such as in bond pricing, generating functions, which can be viewed as a transform of one of the components of the bond price solution, will be constructed. Whilst all of these investigations have direct relevance to quantitative finance and actuarial science, other fields of applications including economics, engineering and other branches of applied sciences could also positively benefit from the results of this research programme.