Stochastic calculus for fractional Lévy processes and related processes

随机过程可以看作是受随机影响的时变系统的模型。对金融、计量经济学、水文学或互联网流量等不同领域的许多现实世界现象的统计分析揭示了长期记忆效应。可以说，研究得最好的具有长期依赖的随机过程是分数布朗运动（Hurst参数H > 1=2）。分数阶lsamvy过程是分数阶布朗运动的概括，它以类似的方式捕捉记忆效应，但在分布建模方面提供了更大的灵活性。该项目的目的是建立分数阶lsamvy过程和相关过程的随机演算，这些过程可以通过确定性核与lsamvy过程的卷积得到。我们将重点关注Skorokhod意义上的随机积分，其零期望性质对于将其解释为可加性噪声模型非常重要。特别是，我们计划研究这些积分的变量变换公式（Itô公式）以及它们在测度变化下的行为。作为一个应用，我们将尝试推导广义分数阶Ornstein-Uhlenbeck过程的积分表示公式，这是金融市场波动的有前途的模型。

Stochastic processes can be considered as models for time dependent systems which are subjectto random infuences. A statistical analysis of many real world phenomena in such diverse fields as finance, econometrics, hydrology, or internet traffic, reveals long memory effects. Arguably, the best-studied stochastic processes with long range dependence are fractional Brownian motions (with Hurst parameter H > 1=2). Fractional Lévy processes are generalizations of fractional Brownian motions which capture the memory effects in a similar fashion but provide more flexibility concerning the modeling of the distribution. The aim of the project is to establish a stochastic calculus for fractional Lévy processes and related processes which can be obtained by a convolution of a deterministic kernel with a Lévy process. We will focus on stochastic integrals in the Skorokhod sense, whose zero expectation property is important for its interpretation as a model for additive noise. In particular, we plan to study the change of variables formula (Itô formula) for these integrals and their behaviour under change of measure. As an application we will try to derive an integral representation formula for generalized fractional Ornstein-Uhlenbeck processes, which are promising models for volatility in financial markets.