Hidden markov models in quantitative finance

我的研究计划包括相关项目，调查市场变量的动态建模问题，这是金融估值，风险管理和资产配置的基本问题。这些项目可分为两大类。第一类项目考虑使用的金融模型，incoporate的经济制度的转变，假设遵循一个隐藏的马尔可夫模型。在此框架下，期望（i）通过求解偏微分方程或常微分方程系统来获得金融合同的公平价格的数值近似，（ii）对保险和养老金相关产品进行估值，其中死亡率和利率都具有随机演变，不一定相互独立，以及（iii）采用Esscher变换或构造其他风险-在马尔可夫市场的不完全设置中执行估值的中性措施。 第二类项目涉及金融模型的使用，其中经济制度的转换由离散或连续时间的隐藏弱马尔可夫链调制。弱马尔可夫链能够捕捉金融过程的动态与记忆。根据这一提法，（i）将采用随机过滤理论来估计模型参数;（ii）估计的参数将用于金融衍生工具定价，计算风险措施和设计最佳资产配置;以及（iii）将解决几个重要的统计推断问题。 解决上述问题将使金融机构能够相应地量化和管理其风险状况，并推动其他研究人员分析金融领域的相关问题。这项研究工作将有助于实现一个高效、运转良好和稳定的金融市场。

My research program comprises of related projects investigating issues on the dynamic modelling of market variables, which is a fundamental concern in financial valuation, risk management and asset allocation. These projects can be grouped into two general categories. The first category of projects considers the use of financial models that incoporate the shift of economic regimes assumed to follow a hidden Markov model. Under this framework, it is desired to (i) obtain numerical approximation for the fair price of financial contracts by solving a system of partial or ordinary differential equations, (ii) value insurance and pension-related products where both mortality and interest rates have stochastic evolution not necessarily independent of each other and (iii) employ Esscher transform or construct other risk-neutral measures to perform valuation in the incomplete setting of a Markov market.  The second category of projects deals with the use of financial models where the switching of economic regimes is modulated by a hidden weak Markov chain in either discrete or continuous time. Weak Markov chains are capable of capturing the dynamics of financial processes with memory. Under this formulation, (i) the theory of stochastic filtering will be adopted to estimate model parameters; (ii) estimated parameters will then be used to price financial derivatives, calculate risk measures and design an optimal allocation of assets; and (iii) several important statistical inference questions will be addressed.  Resolving the aforementioned issues would enable financial institutions to quantify and manage their risk profile accordingly and provide impetus to other researchers in the analysis of related problems in finance. This research work will be instrumental in the attainment of an efficient, well-functioning and stable financial market.