Basic stochastic processes : a course through exercises

書誌事項

Basic stochastic processes : a course through exercises

Zdzisław Brzeźniak and Tomasz Zastawniak

(Springer undergraduate mathematics series)

Springer-Verlag, 2000, c1999

3rd printing with corrections

  • : pbk

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注記

First published: London : Springer-Verlag, c1999

Includes index

内容説明・目次

内容説明

Stochastic processes are tools used widely by statisticians and researchers working in the mathematics of finance. This book for self-study provides a detailed treatment of conditional expectation and probability, a topic that in principle belongs to probability theory, but is essential as a tool for stochastic processes. The book centers on exercises as the main means of explanation.

目次

1. Review of Probability.- 1.1 Events and Probability.- 1.2 Random Variables.- 1.3 Conditional Probability and Independence.- 1.4 Solutions.- 2. Conditional Expectation.- 2.1 Conditioning on an Event.- 2.2 Conditioning on a Discrete Random Variable.- 2.3 Conditioning on an Arbitrary Random Variable.- 2.4 Conditioning on a ?-Field.- 2.5 General Properties.- 2.6 Various Exercises on Conditional Expectation.- 2.7 Solutions.- 3. Martingales in Discrete.- 3.1 Sequences of Random Variables.- 3.2 Filtrations.- 3.3 Martingales.- 3.4 Games of Chance.- 3.5 Stopping Times.- 3.6 Optional Stopping Theorem.- 3.7 Solutions.- 4. Martingale Inequalities and Convergence.- 4.1 Doob's Martingale Inequalities.- 4.2 Doob's Martingale Convergence Theorem.- 4.3 Uniform Integrability and L1 Convergence of Martingales.- 4.4 Solutions.- 5. Markov Chains.- 5.1 First Examples and Definitions.- 5.2 Classification of States.- 5.3 Long-Time Behaviour of Markov Chains: General Case.- 5.4 Long-Time Behaviour of Markov Chains with Finite State Space.- 5.5 Solutions.- 6. Stochastic Processes in Continuous Time.- 6.1 General Notions.- 6.2 Poisson Process.- 6.2.1 Exponential Distribution and Lack of Memory.- 6.2.2 Construction of the Poisson Process.- 6.2.3 Poisson Process Starts from Scratch at Time t.- 6.2.4 Various Exercises on the Poisson Process.- 6.3 Brownian Motion.- 6.3.1 Definition and Basic Properties.- 6.3.2 Increments of Brownian Motion.- 6.3.3 Sample Paths.- 6.3.4 Doob's Maximal L2 Inequality for Brownian Motion.- 6.3.5 Various Exercises on Brownian Motion.- 6.4 Solutions.- 7. Ito Stochastic Calculus.- 7.1 Ito Stochastic Integral: Definition.- 7.2 Examples.- 7.3 Properties of the Stochastic Integral.- 7.4 Stochastic Differential and Ito Formula.- 7.5 Stochastic Differential Equations.- 7.6 Solutions.

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詳細情報

  • NII書誌ID(NCID)
    BA50115584
  • ISBN
    • 3540761756
  • LCCN
    98007021
  • 出版国コード
    gw
  • タイトル言語コード
    eng
  • 本文言語コード
    eng
  • 出版地
    Berlin ; Heidelberg ; New York
  • ページ数/冊数
    x, 225 p.
  • 大きさ
    24 cm
  • 分類
  • 件名
  • 親書誌ID
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