Inference for Stochastic Processes and Applications

波动性是衡量资产价格在一定时期内的预期波动幅度*，波动性越大，风险越高。随机波动率（SV）模型的滤波和递归参数估计在金融决策中有着广泛的应用。SV模型通常用于金融应用，因为它们的动态足够灵活，可以模拟观察到的资产和衍生品价格。*许多应用决策问题，如投资组合选择和期权定价，本质上是递归的。**波动率的推断在期权定价应用中起着重要作用。在基本的布莱克-斯科尔斯-默顿期权定价方法中，波动性是恒定的。在需要建立模型时，日志的*平方值之间的显著相关性超出了这个恒定的波动性。因此，非线性广义自回归条件异方差（GARCH）模型与非线性随机波动模型一起出现了。**从业者在金融和经济学中使用随机系数波动率（RCV）模型。非线性GARCH *模型在许多资产市场的波动动力学建模中非常流行和有效。*我们开发了一种数据驱动的期权定价方法，并用实际数据证明了GARCH/SV *期权定价模型的优越性。**(a)在本研究中，我们研究了GARCH模型、*最近提出的ACP（自回归条件泊松）/RCV模型、持续时间模型、整数值模型、非线性随机波动模型、循环时间序列模型和半鞅模型等随机过程的推理问题。连续时间和离散时间模型的统一函数理论估计方法将用于获得联合最大信息递归估计/过滤估计，并将应用于从期权价格推断和基于审查数据的推断。**(b)对具有无限方差的随机过程的兴趣越来越大，例如Fama *（今年诺贝尔经济学建模奖获得者）研究了无限回归*模型的估计和预测。这是由于非正态稳定定律提供的固有挑战和理论兴趣，以及由这些定律构建的过程可能是许多不同现象的适当模型的可能性。在实践中，任何时间序列，如果表现出尖锐的尖峰或偶尔爆发的孤立*观测值，都表明可能使用具有无限方差的稳定误差模型。对于具有无限方差稳定误差的时间序列*模型，由于无法得到密度的封闭表达式*，因此无法得到最大似然估计。我们用正弦和余弦组合估计函数来研究估计。最近，我开发了一种最大信息量递归方法*，并应用于金融数据。在本提案中，我还将研究使用基于变换的估计函数对无限方差过程进行最大信息滤波/联合递归*估计。(c)实施随机利率模型的问题之一是理论模型价格与现有观察到的债券市场价格不相符。原因是在任何时候都有当前债券价格的*向量，只有几个参数的模型根本无法拟合整个债券价格集。有一个利率模型（随时间变化的参数）可以更准确地拟合观察到的价格是有用的。本文采用非参数估计方法，基于最近提出的具有时变参数的利率模型研究债券价格。

Volatility is a measure of the amount by which an asset price is expected to fluctuate over a given period,*and the greater the volatility, the higher the risk. Filtering and recursive parameter estimation for stochastic*volatility (SV) models have many applications in financial decision making. SV models are commonly used *in financial applications as their dynamics are flexible enough to model observed asset and derivative prices. *Many applied decision making problems such as portfolio selection and option pricing are recursive in nature.**Inference for the volatility plays an important role in option pricing applications. Constant volatility has been *assumed in the basic Black-Scholes-Merton approach to option pricing. Significant correlation among the *squared values of the log returns points at a need to model beyond this constant volatility. As a consequence, *the world of nonlinear generalized autoregressive conditional heterocedastic (GARCH) modeling together *with nonlinear stochastic volatility models has emerged.**Practitioners use random coefficient volatility (RCV) models in finance and economics. Nonlinear GARCH *models have been very popular and effective for modeling volatility dynamics in many asset markets. *We have developed a data driven method for option pricing and demonstrated the superiority of GARCH/SV *option pricing models using real data.**(a) In this proposed research, we study inference problems for stochastic processes such as GARCH models,*recently proposed ACP ( autoregressive conditionally Poisson)/RCV models, duration models, integer *valued models, nonlinear stochastic volatility models, circular time series models and semimartingale models.*The unified method of estimating function theory for continuous time as well as for discrete time models will*be used to obtain joint maximum informative recursive estimates/filtered estimates and will be applied to *inferences from option prices and to inference based on censored data.**(b) There has been a growing interest in stochastic processes with infinite variance, for example Fama *(Nobel Price Winner for econometric modelling this year) studied estimation and prediction for infinite regression*models. This is due to the inherent challenge and theoretical interest provided by the non-normal stable laws as *well as the possibility that the processes constructed from these laws may be appropriate models for many *diverse phenomena. In practice, any time series which exhibits sharp spikes or occasional bursts of outlying *observations suggests the possible use of a model with stable errors having infinite variance. For time series *models with infinite variance stable errors, for which closed form expressions for the density are not available *and hence the maximum likelihood estimate cannot be obtained.We have used combined sine and cosine *estimating functions to study estimation. Recently I developed a maximum informative recursive method *and applied to financial data. In this proposal, I will also study maximum informative filtering/joint recursive *estimation for infinite variance processes using transformation based estimating functions.**(c) One of the problems with the implementations of stochastic interest rate models was that the theoretical *model prices did not fit the existing observed market prices of bonds. The reason is that at any time there is a*vector of current bond prices, and a model with a few parameters simply cannot fit the entire set of bond prices.*It is useful to have an interest rate model (with time varying parameters) which can fit the observed prices more*accurately. We use the nonparametric estimation method to study bond prices based on recently proposed *interest rate models with time varying parameters.