Encyclopaedia of Mathematical Sciences

Stochastic Calculus

By: S.V. Anulova (Contribution by), Yurij V. Prokhorov (Editor), P.B. Slater (Translator)

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Hardcover | 25 November 1997

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270 Pages

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Preface In the axioms of probability theory proposed by Kolmogorov the basic "probabilistic" object is the concept of a probability model or probability space. This is a triple (n, F, P), where n is the space of elementary events or outcomes, F is a a-algebra of subsets of n announced by the events and P is a probability measure or a probability on the measure space (n, F). This generally accepted system of axioms of probability theory proved to be so successful that, apart from its simplicity, it enabled one to embrace the classical branches of probability theory and, at the same time, it paved the way for the development of new chapters in it, in particular, the theory of random (or stochastic) processes. In the theory of random processes, various classes of processes have been studied in depth. Theories of processes with independent increments, Markov processes, stationary processes, among others, have been constructed. In the formation and development of the theory of random processes, a significant event was the realization that the construction of a "general theory of ran- dom processes" requires the introduction of a flow of a-algebras (a filtration) F = (Ftk:: o supplementing the triple (n, F, P), where F is interpreted as t the collection of events from F observable up to time t.

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You Can Find This Book In

Non-Fiction Mathematics Probability & Statistics Engineering & Technology Electronics & Communications Engineering Electronics Engineering Accounting & Finance Finance Applied Mathematics Stochastics Calculus & Mathematical Analysis Calculus

Technology in General Engineering in General Computing & I.T.Computer Science Artificial Intelligence

Preface	p. 5
Introduction to Stochastic Calculus	p. 7
Stochastic Differential and Evolution Equations	p. 38
Stochastic Differential Equations (SDEs)	p. 38
Stochastic Evolution Equations	p. 76
Stochastic Calculus (Malliavin Calculus). Applications to Stochastic Differential Equations	p. 91
Stochastic Calculus on Filtered Probability Spaces	p. 111
Elements of the General Theory of Stochastic Processes	p. 111
Semimartingales. Stochastic Integrals	p. 125
Absolute Continuity and Singularity of Probability Distributions	p. 140
Martingales and Limit Theorems for Stochastic Processes	p. 158
Theory: Weak Convergence of Probability Measures on Metric Spaces	p. 158
Applications: The Invariance Principle and Diffusion Approximation	p. 213
Author Index	p. 249
Subject Index	p. 251
Table of Contents provided by Blackwell. All Rights Reserved.

Encyclopaedia of Mathematical Sciences

Stochastic Calculus

At a Glance

Hardcover

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