Lecture Notes in Mathematics

A Stochastic Calculus Approach

By: Sigurd Assing, Wolfgang M. Schmidt

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Paperback | 20 May 1998

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152 Pages

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The book presents an in-depth study of arbitrary one-dimensional continuous strong Markov processes using methods of stochastic calculus. Departing from the classical approaches, a unified investigation of regular as well as arbitrary non-regular diffusions is provided. A general construction method for such processes, based on a generalization of the concept of a perfect additive functional, is developed. The intrinsic decomposition of a continuous strong Markov semimartingale is discovered. The book also investigates relations to stochastic differential equations and fundamental examples of irregular diffusions.

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Preface
Basic Concepts and Preparatory Results	p. 1
Basic Probability Structure	p. 1
Strong Markov Processes	p. 2
Semimartingales	p. 3
Local Times and the Generalized Ito-formula	p. 5
Random Time Change	p. 10
Equality of Distributions	p. 11
Extension of the Basic Probability Structure	p. 12
Classification of the Points of the State Space	p. 15
Classification of the Points	p. 15
Decomposition of the State Space	p. 16
Auxiliary Results	p. 19
Weakly Additive Functionals and Time Change of Strong Markov Processes	p. 27
Semimartingale Decomposition of Continuous Strong Markov Semimartingales	p. 33
Continuous Strong Markov Processes with Increasing Paths	p. 33
Intrinsic Structure of the Finite Variation Part in the Semimartingale Decomposition	p. 34
Occupation Time Formula	p. 53
Speed Measure	p. 53
Occupation Time Formula	p. 62
Construction of Continuous Strong Markov Processes	p. 79
Continuous Strong Markov Semimartingales as Solutions of Stochastic Differential Equations	p. 103
Equations with Generalized Drift	p. 103
Equations with Ordinary Drift	p. 107
Fundamental Examples of Non-regular Diffusions	p. 112
App. 1	p. 119
App. 2	p. 127
References	p. 133
Index	p. 136
Symbols	p. 137
Table of Contents provided by Blackwell. All Rights Reserved.

Lecture Notes in Mathematics

A Stochastic Calculus Approach

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Paperback

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