Lévy–Driven Continuous–Time ARMA Processes
Lévy–Driven Continuous–Time ARMA Processes
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Lévy 驱动的连续时间 ARMA 过程
DOI：
10.1007/978-3-540-71297-8_20
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
P. Brockwell
中科院分区：
文献类型：
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作者：
P. Brockwell
具有连续时间参数的高斯阿尔马过程，也称为具有有理谱密度的平稳连续时间高斯过程，多年来一直受到人们的关注。(See例如Doob（1944）、Bartlett（1946）、菲利普斯（1959）、Durbin（1961）、Dzhapararidze（1970，1971）、Pham-Din-Tuan（1977）的论文和Arato（1982）的专著。在过去的20年里，人们对连续时间过程的兴趣又重新兴起，部分原因是随机微分方程模型在金融问题中的成功应用，布莱克-斯科尔斯期权定价公式的推导及其推广就是例证（船体和白色（1987））。连续时间模型的计量经济学应用的许多例子包含在Bergstrom（1990）的书中。连续时间模型也被非常成功地用于不规则间隔数据的建模（Jones（1981，1985），Jones and Ackerson（1990））。连续时间阿尔马过程与离散时间过程一样，是一个非常方便的平稳过程参数族，具有广泛的自相关函数，可用于模拟金融时间序列分析中的经验自相关性。在金融应用中，人们已经注意到，跳跃在资产价格和衍生系列（如波动率）的现实建模中发挥着重要作用。这导致了对Levy过程及其在金融建模中的应用的兴趣激增。在这篇文章中，我们讨论了二阶列维驱动的连续时间阿尔马模型，他们的性质和他们的一些金融应用。例子是Barndorff-Nielsen和Shephard（2001）引入的一类模型中的随机波动率建模，以及一类连续时间Gestival模型的构建，该模型推广了Kluppelberg，Lindner和Maller（2004）的COGestival（1，1）过程，并表现出类似于离散时间Gestival（p，q）过程的性质。
Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density, have been of interest for many years. (See for example the papers of Doob (1944), Bartlett (1946), Phillips (1959), Durbin (1961), Dzhapararidze (1970,1971), Pham-Din-Tuan (1977) and the monograph of Arato (1982).) In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations (Hull and White (1987)). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990). Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data (Jones (1981, 1985), Jones and Ackerson (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary processes exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in financial time series analysis. In financial applications it has been observed that jumps play an important role in the realistic modelling of asset prices and derived series such as volatility. This has led to an upsurge of interest in Levy processes and their applications to financial modelling. In this article we discuss second-order Levy-driven continuous-time ARMA models, their properties and some of their financial applications. Examples are the modelling of stochastic volatility in the class of models introduced by Barndorff-Nielsen and Shephard (2001) and the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Kluppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discretetime GARCH(p,q) process.