Stochastic integration with jumps / Klaus Bichteler.

Format:

Book

Author/Creator:

Bichteler, Klaus, author.

Series:

Encyclopedia of mathematics and its applications ; v. 89.

Encyclopedia of mathematics and its applications ; volume 89

Language:

English

Subjects (All):

Stochastic integrals.

Jump processes.

Physical Description:

1 online resource (xiii, 501 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2002.

Language Note:

English

Summary:

Stochastic processes with jumps and random measures are importance as drivers in applications like financial mathematics and signal processing. This 2002 text 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 results from ordinary integration theory, for instance previsible envelopes and an algorithm computing stochastic integrals of càglàd integrands pathwise. Full proofs are given for all results, and motivation is stressed throughout. A large appendix contains most of the analysis that readers will need as a prerequisite. This will be an invaluable reference for graduate students and researchers in mathematics, physics, electrical engineering and finance who need to use stochastic differential equations.

Contents:

Motivation: Stochastic Differential Equations

Wiener Process

The General Model

Integrators and Martingales

The Elementary Stochastic Integral

The Semivariations

Path Regularity of Integrators

Processes of Finite Variation

Martingales

Extension of the Integral

The Daniell Mean

The Integration Theory of a Mean

Countable Additivity in p-Mean

Measurability

Predictable and Previsible Processes

Special Properties of Daniell's Mean

The Indefinite Integral

Functions of Integrators

Ito's Formula

Random Measures

Control of Integral and Integrator

Change of Measure

Factorization

Martingale Inequalities

The Doob-Meyer Decomposition

Semimartingales

Previsible Control of Integrators

Levy Processes

Stochastic Differential Equations

Existence and Uniqueness of the Solution

Stability: Differentiability in Parameters

Pathwise Computation of the Solution

Weak Solutions

Stochastic Flows

Semigroups, Markov Processes, and PDE

Complements to Topology and Measure Theory

Notations and Conventions

Topological Miscellanea

Measure and Integration

Weak Convergence of Measures

Analytic Sets and Capacity

Suslin Spaces and Tightness of Measures

The Skorohod Topology

The L[superscript p]-Spaces

Semigroups of Operators.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

ISBN:

1-139-88299-6

1-107-10144-1

1-107-10396-7

0-521-14214-8

0-511-54987-3

1-107-09586-7

0-511-02073-2

1-107-09269-8

OCLC:

668203477