Lévy過程

前言

　　Lévy processes can be thought of as random walks in continuous time，that is they are stochastic processes with independent and stationary increments．The state space may be a fairly general topological group，but in this text．we will stick to the Euclidean framework．The best known and most important examples are the Poisson process，Brownian motion，the Cauchy process，and more generally stable processes．Lbvy processes concern many aspects of probability theory and its applications．In particular，they are prototypes of Markov processes(actually，they　　form the class of space-time homogeneous Markov processesl and of semimartingales：they are also used as models in the study of queues．insurance risks，dams，and more recently in mathematical finance．From the viewpoint of functional analysis．they appear in connection with potential theory of convolution semigroups．　　Historically，the first researches go back to the late 20’S fthat iS when the foundations of modern probability theory were laid down)with the study of infinitely divisible distributions．Their general structure has been gradually discovered by de Finetti，Koimogorov，L6vy，Khintchine and It6：it iS described by the celebrated L6vy．Khintchine formula which points out the correspondence between infinitely divisible distributions and processes with independent and stationary increments．Atier the pioneer contribution of Hunt in the mid-50s，the spreading of the theory of Markov processes and its connection with abstract potential theory has had a considerable impact on L6vy processes；see the works of Doob，Dynkin，Blumenthal and Getoor，Skorohod，Kesten，Bretagnolle．

內(nèi)容概要

本書是一部全新、綜合描述Levy過程理論的教程。近年來，Levy過程理論作為現(xiàn)代概率的重要一支得到了迅速的發(fā)展，在序列、數(shù)學(xué)金融和風(fēng)險(xiǎn)估計(jì)等各個(gè)領(lǐng)域的應(yīng)用廣泛。Bertoin教授運(yùn)用概率結(jié)構(gòu)和分析工具之間強(qiáng)有力的聯(lián)系將這個(gè)核心理論講述的相當(dāng)簡(jiǎn)明。介紹從屬過程的特殊性質(zhì)以及其在研究實(shí)值Levy過程和起伏理論時(shí)的關(guān)鍵特征。詳盡講述了沒有正跳躍的Levy過程和平穩(wěn)過程。目次：基礎(chǔ)；馬爾科夫過程的Levy過程；勢(shì)理論基礎(chǔ)；局部時(shí)間和馬爾科夫游弋；Levy過程的局部時(shí)間；起伏理論；沒有正跳躍的Levy過程；平穩(wěn)過程和標(biāo)度特征。    讀者對(duì)象：本書適用于所有對(duì)概率論感興趣的科研人員。

書籍目錄

Preface0   Preliminaries  1 Notation  2 Infinitely divisible distributions  3 Martingales  4 Poisson processes  5 Poisson measures and Poisson point processes  6 Brownian motion  7 Regular variation and Tauberian theoremsI    Levy Processes as Markov Processes  1 Levy processes and the Lbvy-Khintchine formula  2 Markov property and related operators  3 Absolutely continuous resoivents  4 Transience and recurrence  5 Exercises  6 CommentsII   Elements of Potential Theory  1 Duality and time reversal  2 Capacitary measure  3 Essentially polar sets and capacity  4 Energy  5 The case of a single point  6 Exercises  7 CommentsIII  Subordimtors  1 Definitions and first properties  2 Passage across a level  3 The arcsine laws  4 Rates of growth  5 Dimension of the range  6 Exercises  7 CommentsIV  Local Time and Excursions of a Markov Process  1 Framework  2 Construction of the local time  3 Inverse local time  4 Excursion measure and excursion process  5 The cases of holding points and of irregular points  6 Exercises   7 CommentsV   Local Times of a Levy Process  1 Occupation measure and local times  2 Hilbert transform of local times  3 Jointly continuous local times  4 Exercises  5 CommentsVI  Fluctuation Theory  1 The reflected process and the ladder process  2 Fluctuation identities  3 Some applications of the ladder time process  4 Some applications of the ladder height process  5 Increase times  6 Exercises  7 CommentsVll  Levy Processes with no Positive Jumps  1 Fluctuation theory with no positive jumps  2 The scale function  3 The process conditioned to stay positive  4 Some path transformations  5 Exercises  6 CommentsVIII Stable Processes and the Scaling Property  1 Definition and probability estimates  2 Some sample path properties  3 Bridges  4 Normalized excursion and meander  5 Exercises  6 CommentsReferencesList of symbolsIndex

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