# Foundations of modern probability

@inproceedings{Kallenberg1997FoundationsOM, title={Foundations of modern probability}, author={Olav Kallenberg}, year={1997} }

* 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…

## 3,939 Citations

Canonical Sample Spaces for Random Dynamical Systems

- Mathematics
- 2009

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

- Mathematics
- 2007

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

- Computer Science, MathematicsJ. Appl. Probab.
- 2015

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

- MathematicsJournal of Applied Probability
- 2015

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

- Mathematics
- 2017

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

- Mathematics
- 2014

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

- Computer Science
- 2016

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

- Mathematics
- 2012

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

- Mathematics
- 2019

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

- 2007

Some fundamental properties related to Lévy processes are discussed. Topics include infinitely divisible distributions, Lévy-Khintchine formula, Poisson random measures, Lévy-Itô decomposition,…

## References

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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

- Mathematics
- 1987

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

- Mathematics
- 1992

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

- Mathematics
- 1990

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

- Mathematics
- 1984

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

- Mathematics
- 1978

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…

STOCHASTIC INTEGRAL EQUATIONS

- Mathematics
- 1980

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

- Mathematics
- 1980

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

- Mathematics, Computer Science
- 1987

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

- Mathematics
- 1968

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…