《Conditional Distributions of Mandelbrot–van ness Fractional LÉVY Processes and Continuous-Time ARMA–GARCH-Type Models with Long Memory》

此篇论文发表在2016年JOURNAL OF TIME SERIES ANALYSIS上。 作者在文中提出把Mandelbrot–Van Ness fractional Lévy processes 应用到连续时间ARMA–GARCH模型中。

简介

Long-memory effects can be found in many data sets from finance to hydrology. Therefore, models that can reflect these properties have become more popular in recent years. Mandelbrot–Van Ness fractional Lévy processes allow for such stationary long-memory effects in their increments and have been used in different settings ranging from fractionally integrated continuous-time ARMA–GARCH-type setups to general stochastic differential equations. However, their conditional distributions have not yet been considered in detail. In this article, we provide a closed formula for their conditional characteristic functions and suggest several applications to continuous-time ARMA–GARCH-type models with long memory.

导论

Fractional Lévy processes can be obtained in several ways via convolution of classical Lévy processes and have been introduced to allow for long-memory effects in increments. There exist several possibilities to choose these convolution integrands with the Mandelbrot–Van Ness and Molchan–Golosov kernels being the most prominent examples. However, in contrast to the Brownian motion, both of these approaches do not lead to the same kind of fractional processes. The Mandelbrot–Van Ness fractional Lévy processes (MvN-fLps), which have been considered by Marquardt (2006), allow for stationary increments, while the Molchan–Golosov fractional Lévy processes (MG-fLps) offer the possibility of having fractional subordinators, that is, strictly increasing processes (cf. Bender and Marquardt (2009), Tikanmäki and Mishura (2011) or Fink (2013)). While the later ones could be a good choice for derivative pricing in, for example, interest rate, credit or stochastic volatility settings where positive processes are needed, the first ones might be better suited for time series analysis because of their stationarity (e.g. Haug and Czado, 2007). In all these situations, however, whether one is interested in time series prediction or derivative pricing, it is very useful to understand the respective conditional distributions. While these have been considered already by Fink (2013) for MG-fLps, to the best of our knowledge, there is no full characterization yet for the MvN-fLps in the literature. In this article, we want to close this gap by providing a closed formula for the conditional characteristic function of these processes, which completely characterizes the conditional distribution and allows for easy and quick calculations of moments like conditional expectation and conditional variance via differentiation.