Inequalities in Analysis and Probability

The book is aimed at graduate students and researchers with basic knowledge of Probability and Integration Theory. It introduces classical inequalities in vector and functional spaces with applications to probability. It also develops new extensions of the analytical inequalities, with sharper bounds and generalizations to the sum or the supremum of random variables, to martingales and to transformed Brownian motions. The proofs of many new results are presented in great detail. Original tools are developed for spatial point processes and stochastic integration with respect to local martingales in the plane.

This second edition covers properties of random variables and time continuous local martingales with a discontinuous predictable compensator, with exponential inequalities and new inequalities for their maximum variable and their p-variations. A chapter on stochastic calculus presents the exponential sub-martingales developed for stationary processes and their properties. Another chapter devoted itself to the renewal theory of processes and to semi-Markovian processes, branching processes and shock processes. The Chapman–Kolmogorov equations for strong semi-Markovian processes provide equations for their hitting times in a functional setting which extends the exponential properties of the Markovian processes.

Contents:

• Preliminaries
• Inequalities for Means and Integrals
• Analytic Inequalities
• Inequalities for Martingales
• Stochastic Calculus
• Functional Inequalities
• Markov Processes
• Inequalities for Processes
• Inequalities in Complex Spaces
• Appendix A: Probability

Readership: Graduate students and researchers in probability and integration theory.
Key Features:

• It provides a unique treatment for stochastic processes and equations of their hitting times
• The additional chapters present new results which generalize the existing theory of stochastic processes and calculus
• It covers functional framework for the renewal theory, the exponential sub-martingales and the hitting times and the local times of Gaussian processes and processes with independent increments
• The properties of the hitting times of the semi-Markovian processes are presented in an original functional framework