Stochastic Calculus for Fractional Brownian Motion and Applications
Stochastic Calculus for Fractional Brownian Motion and Applications
复制标题
DOI：
10.1007/978-1-84628-797-8
复制
发表时间：
2008-02
期刊：
影响因子：
--
通讯作者：
F. Biagini;Yaozhong Hu;B. Øksendal;Tusheng Zhang
中科院分区：
文献类型：
--
作者：
F. Biagini;Yaozhong Hu;B. Øksendal;Tusheng Zhang
分数布朗运动(FBM)已被广泛应用于从生物学到金融等不同领域的许多现象的建模。这种巨大的潜在应用范围使FBM成为一个有趣的研究对象。FBM表示经典布朗运动的一个自然的单参数推广，因此人们很自然地会问是否可以建立一个关于FBM的随机演算。这一点并不明显，因为FBM既不是半鞅(H=1⁄2时除外)，也不是马尔可夫过程，所以经典的随机演算的数学机制在FBM的情况下是不可用的。有几种方法被用来发展FBM的随机微积分的概念。这本书的目的是全面介绍关于FBM的随机积分的不同定义，并给出由此产生的理论的应用。特别强调研究不同方法之间的关系。读者被认为熟悉概率论和随机分析，尽管书中使用的数学技术已经完全暴露，一些必要的先决条件，如经典的白噪声理论和分数阶微积分，在附录中被回忆起来。这本书将是一个有价值的参考研究生和研究人员在数学，生物，气象，物理，工程和金融。这本书的各个方面也将在其他领域有用，其中FBM可以用作应用程序的模型。
Fractional Brownian motion (fBm) has been widely used to model a number of phenomena in diverse fields from biology to finance. This huge range of potential applications makes fBm an interesting object of study. fBm represents a natural one-parameter extension of classical Brownian motion therefore it is natural to ask if a stochastic calculus for fBm can be developed. This is not obvious, since fBm is neither a semimartingale (except when H= 1⁄ 2), nor a Markov process so the classical mathematical machineries for stochastic calculus are not available in the fBm case. Several approaches have been used to develop the concept of stochastic calculus for fBm. The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance. Aspects of the book will also be useful in other fields where fBm can be used as a model for applications.