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Stochastic integration with jumps 1st Edition by Klaus Bichteler ISBN 0521811295 9780521811293

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Author(s): Klaus Bichteler
ISBN(s): 9780521811293, 0521811295
Edition: 1
File Details: PDF, 3.18 MB
Year: 2002
Language: english
SKU: EB-897320 Category: Tag:

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Product details:

ISBN 10: 0521811295 
ISBN 13: 9780521811293
Author: Klaus Bichteler

Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. This book develops stochastic integration theory for both integrators (semimartingales) and random measures from a common point of view. Using some novel predictable controlling devices, the author furnishes the theory of stochastic differential equations driven by them, as well as their stability and numerical approximation theories. Highlights feature DCT and Egoroff’s Theorem, as well as comprehensive analogs to results from ordinary integration theory, for instance, previsible envelopes and an algorithm computing stochastic integrals of c`agl`ad integrands pathwise.

Stochastic integration with jumps 1st Table of contents:

  • Chapter 1: Introduction

    • Motivation: Stochastic Differential Equations
    • Wiener Process
    • Brownian Motion
    • Stochastic Integrals
    • Probabilities
    • The Sizes of Random Variables
    • Two Notions of Equality for Processes
    • The Natural Conditions

  • Chapter 2: Integrators and Martingales

    • Step Functions and Lebesgue-Stieltjes Integrators on the Line
    • The Elementary Stochastic Integral
    • The Semivariations
    • The Process (Xt)t∈[0,∞)
    • Processes of Finite Variation
    • Martingales

  • Chapter 3: Extension of the Integral

    • The Space L(X,P)
    • The Integration Theory of Vectors of Integrators
    • Permanence Under Limits of Sequences
    • Permanence Under Algebraic and Linear Operations
    • Predictable Processes
    • Previsible Processes
    • Predictable Envelopes
    • The Indefinite Integral
    • Functions of Integrators
    • Ito’s Formula

  • Chapter 4: Control of Integral and Integrator

    • σ-Additivity
    • Law and Canonical Representation
    • Example: Wiener Process
    • The Dual Predictable Projection
    • Fefferman’s Inequality
    • The Burkholder-Davis-Gundy Inequalities
    • The Doleans-Dade Measures and Processes
    • Integrators Are Semimartingales
    • Various Decompositions of an Integrator
    • Controlling a Single Integrator
    • Previsible Control of Vectors of Integrators
    • The Levy-Khintchine Formula
    • The Martingale Representation Theorem

  • Chapter 5: Stochastic Differential Equations

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Tags: Klaus Bichteler, Stochastic, integration