Martingales and Markov Chains: Solved Exercises and Elements of Theory

by Baldi; Paolo

• ISBN13: 9781584883296
• ISBN10: 1584883294
• Format: Nonspecific Binding
• Copyright: 2002-04-26
• Publisher: Chapman & Hall/

Summary

A thorough grounding in Markov chains and martingales is essential in dealing with many problems in applied probability, and is a gateway to the more complex situations encountered in the study of stochastic processes. Exercises are a fundamental and valuable training tool that deepen students' understanding of theoretical principles and prepare them to tackle real problems.In addition to a quick but thorough exposition of the theory, Martingales and Markov Chains: Solved Exercises and Elements of Theory presents, more than 100 exercises related to martingales and Markov chains with a countable state space, each with a full and detailed solution. The authors begin with a review of the basic notions of conditional expectations and stochastic processes, then set the stage for each set of exercises by recalling the relevant elements of the theory. The exercises range in difficulty from the elementary, requiring use of the basic theory, to the more advanced, which challenge the reader's initiative. Each section also contains a set of problems that open the door to specific applications.Designed for senior undergraduate- and graduate level students, this text goes well beyond merely offering hints for solving the exercises, but it is much more than just a solutions manual. Within its solutions, it provides frequent references to the relevant theory, proposes alternative ways of approaching the problem, and discusses and compares the arguments involved.

Author Biography

Paolo Baldi is a Professor in the Department of Mathematics at the Universita di Roma "Tor Vergata", Italy. Laurent Mazliak and Pierre Priouret are Professors in the Laboratoire de Probabilites et Modeles Aleatoires at the Universite de Paris 6, France

Table of Contents

CONDITIONAL EXPECTATIONS

Introduction

Definition and First Properties

Conditional Expectations and Conditional Laws

Exercises

Solutions

STOCHASTIC PROCESSES

General Facts

Stopping Times

Exercises

Solutions

MARTINGALES

First Definitions

First Properties

The Stopping Theorem

Maximal Inequalities

Square Integral Martingales

Convergence Theorems

Regular Martingales

Exercises

Problems

Solutions

MARKOV CHAINS

Transition Matrices, Markov Chains

Construction and Existence

Computations on the Canonical Chain

Potential Operators

Passage Problems

Recurrence, Transience

Recurrent Irreducible Chains

Periodicity

Exercises

Problems

Solutions

REFERENCES

INDEX