Martingales and Markov Chains: Solved Exercises and Elements of Theory

Paolo Baldi, Laurent Mazliak, Pierre Priouret



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.

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

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