Nicolas Privault is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. He has authored the book, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Mathematics, Springer, 2009 and was a co-editor for the book, Stochastic Analysis with Financial Applications, Progress in Probability, Vol. 65, Springer Basel, 2011. Aside from these two Springer titles, he has authored several others. He is currently teaching the course M27004-Probability Theory and Stochastic Processes at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.

Introduction

1 Probability Background

1.1 Probability Spaces and Events

1.2 Probability Measures

1.3 Conditional Probabilities and Independence

1.4 Random Variables

1.5 Probability Distributions

1.6 Expectation of a Random Variable

1.7 Conditional Expectation

1.8 Moment and Probability Generating Functions

Exercises

2 Gambling Problems

2.1 Constrained Random Walk

2.2 Ruin Probabilities

2.3 Mean Game Duration

Exercises

3 Random Walk

3.1 Unrestricted Random Walk

3.2 Mean and Variance

3.3 Distribution

3.4 First Return to Zero

Exercises

4 Discrete-Time Markov Chains

4.1 Markov Property

4.2 Transition matrix

4.3 Examples of Markov Chains

4.4 Higher Order Transition Probabilities

4.5 The Two-State Discrete-Time Markov Chain

Exercises

5 First Step Analysis

5.1 Hitting Probabilities

5.2 Mean Hitting and Absorption Times

5.3 First Return Times

5.4 Number of Returns

Exercises

6 Classication of States

6.1 Communicating States

6.2 Recurrent States

6.3 Transient States

6.4 Positive and Null Recurrence

6.5 Periodicity and Aperiodicity

Exercises

7 Long-Run Behavior of Markov Chains

7.1 Limiting Distributions

7.2 Stationary Distributions

7.3 Markov Chain Monte Carlo

Exercises

8 Branching Processes

8.1 Defnition and Examples

8.2 Probability Generating Functions

8.3 Extinction Probabilities

Exercises

9 Continuous-Time Markov Chains

9.1 The Poisson Process

9.2 Continuous-Time Chains

9.3 Transition Semigroup9.4 Infinitesimal Generator

9.5 The Two-State Continuous-Time Markov Chain

9.6 Limiting and Stationary Distributions

9.7 The Discrete-Time Embedded Chain

9.8 Mean Absorption Time and Probabilities

Exercises

10 Discrete-Time Martingales

10.1 Filtrations and Conditional Expectations

10.2 Martingales - Definition and Properties

10.3 Ruin Probabilities

10.4 Mean Game Duration

Exercises

11 Spatial Poisson Processes

11.1 Spatial Poisson (1781-1840) Processes

11.2 Poisson Stochastic Integrals

11.3 Transformations of Poisson Measures

11.4 Moments of Poisson Stochastic Integrals

11.5 Deviation Inequalities

Exercises

12 Reliability Theory

12.1 Survival Probabilities

12.2 Poisson Process with Time-Dependent

Intensity

12.3 Mean Time to Failure

Exercises

Some Useful Identities

Solutions to the Exercises

References

Index