Foundations of modern probability

  title={Foundations of modern probability},
  author={Olav Kallenberg},
* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian… 
Canonical Sample Spaces for Random Dynamical Systems
This is an overview about natural sample spaces for differential equations driven by various noises. Appropriate sample spaces are needed in order to facilitate a random dynamical systems approach
Poisson-type deviation inequalities for curved continuous-time Markov chains
In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-curvature is bounded below. Although these two curvatures
Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications
A new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets, which is significantly more efficient than standard Monte Carlo methods.
Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications
We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method
Stochastic partial differential equations driven by Lévy white noises: Generalized random processes, random field solutions and regularity
We study various aspects of stochastic partial differential equations driven by Levy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a
Lévy processes and filtering theory
Stochastic filtering theory is the estimation of a continuous random system given a sequence of partial noisy observations, and is of use in many different financial and scientific areas. The main
General Theory of Markov Processes
This chapter discusses the strong Markov property, and presents three important classes of Markov processes, including Feller processes, which are a fundamental class of stochastic processes with many applications in real life problems outside mathematics.
Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips
We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for
Random Function Iterations for Stochastic Feasibility Problems
The aim of this thesis is to develop a theory that describes errors in fixed point iterations stochastically, treating the iterations as a Markov chain and analyzing them for convergence in
Brief tutorial of Lévy processes
Some fundamental properties related to Lévy processes are discussed. Topics include infinitely divisible distributions, Lévy-Khintchine formula, Poisson random measures, Lévy-Itô decomposition,


Statistics of random processes
1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process,
Limit Theorems for Stochastic Processes
I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems
Random Series and Stochastic Integrals: Single and Multiple
0 Preliminaries.- 0.1 Topology and measures.- 0.2 Tail inequalities.- 0.3 Filtrations and stopping times.- 0.4 Extensions of probability spaces.- 0.5 Bernoulli and canonical Gaussian and ?-stable
Continuous martingales and Brownian motion
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-
Convergence of stochastic processes
I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct
Probability theory: Independence, interchangeability, martingales
1 Classes of Sets, Measures, and Probability Spaces.- 1.1 Sets and set operations.- 1.2 Spaces and indicators.- 1.3 Sigma-algebras, measurable spaces, and product spaces.- 1.4 Measurable
This chapter focuses on stochastic integral equations. It presents classical examples and a theory for general stochastic integral equations with a Lipschitz-type hypothesis. In these equations, the
Stochastic Filtering Theory
1 Stochastic Processes: Basic Concepts and Definitions.- 2 Martingales and the Wiener Process.- 3 Stochastic Integrals.- 4 The Ito Formula.- 5 Stochastic Differential Equations.- 6 Functionals of a
Brownian Motion and Stochastic Calculus
This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.
The Ergodic Theory of Subadditive Stochastic Processes
SUMMARY An ergodic theory is developed for the subadditive processes introduced by Hammersley and Welsh (1965) in their study of percolation theory. This is a complete generalization of the classical