Anteneodo C, Riera R
Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro, CP 38071, 22452-970, Rio de Janeiro, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Aug;72(2 Pt 2):026106. doi: 10.1103/PhysRevE.72.026106. Epub 2005 Aug 3.
We investigate a generalized stochastic model with the property known as mean reversion, that is, the tendency to relax towards a historical reference level. Besides this property, the dynamics is driven by multiplicative and additive Wiener processes. While the former is modulated by the internal behavior of the system, the latter is purely exogenous. We focus on the stochastic dynamics of volatilities, but our model may also be suitable for other financial random variables exhibiting the mean reversion property. The generalized model contains, as particular cases, many early approaches in the literature of volatilities or, more generally, of mean-reverting financial processes. We analyze the long-time probability density function associated to the model defined through an Itô-Langevin equation. We obtain a rich spectrum of shapes for the probability function according to the model parameters. We show that additive-multiplicative processes provide realistic models to describe empirical distributions, for the whole range of data.
我们研究了一个具有均值回复特性的广义随机模型，即趋向于向历史参考水平松弛的趋势。除了该特性外，动力学由乘性和加性维纳过程驱动。前者由系统的内部行为调制，而后者纯粹是外生的。我们关注波动率的随机动力学，但我们的模型也可能适用于其他表现出均值回复特性的金融随机变量。作为特殊情况，广义模型包含了文献中许多关于波动率或更一般地关于均值回复金融过程的早期方法。我们分析了通过伊藤 - 朗之万方程定义的模型相关的长期概率密度函数。根据模型参数，我们得到了概率函数的丰富形状谱。我们表明，对于整个数据范围，加性 - 乘性过程提供了描述经验分布的现实模型。
