Probability Theory III

1. Front Matter

   Pages i-6

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2. Introduction to Stochastic Calculus

   • N. V. Krylov

   Pages 7-37

3. Stochastic Differential and Evolution Equations

   • S. V. Anulova, A. Yu. Veretennikov

   Pages 38-110

4. Stochastic Calculus on Filtered Probability Spaces

   • R. Sh. Liptser, A. N. Shiryaev

   Pages 111-157

5. Martingales and Limit Theorems for Stochastic Processes

   • R. Sh. Liptser, A. N. Shiryaev

   Pages 158-247

6. Back Matter

   Pages 249-256

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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.

• Brownian motion
• Brownsche Bewegung
• Konvergenz von Prozessen
• Malliavin Kalkül
• Malliavin calculus
• Martingale
• Probability space
• Probability theory
• Stochastisches Integral
• calculus
• differential equation
• mathematical statistics
• semimartingale
• statistics
• stochastic integral

• Steklov Mathematical Institute, Moscow, Russia

  Yu. V. Prokhorov, A. N. Shiryaev

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