This is the expanded second edition of a successful textbook that provides a broad introduction to the important area of stochastic modelling. The original text had been developed from lecture notes for a one-semester course on the topic for third-year science and actuarial students at the University of Melbourne. It reviews the basics of probability theory, and then covers the following topics: Markov chains, Markov decision processes, jump Markov processes, elements of queueing theory, basic renewal theory, elements of time series and simulation. The present edition adds new chapters on elements of stochastic calculus and introductory mathematical finance that logically complement the topics chosen for the first edition. This makes the book suitable for a larger variety of university courses presenting the fundamentals of modern stochastic modelling. Rigorous proofs are often replaced with sketches of arguments - with indications as to why a particular result holds, and also how it is connected to other results - and illustrated by well-selected examples.Wherever possible, the book includes references to more specialised texts containing both proofs and more advanced material related to the topics covered.
Preface to the First Edition vii
Preface to the Second Edition xi
1 Introduction 1 (8)
2 Basics of Probability Theory 9 (68)
2.1 Probability Spaces 10 (10)
2.2 Distributions and Integrals 20 (6)
2.3 Conditional Probability and Independence 26 (2)
2.4 Random Variables and Their Distributions 28 (8)
2.5 Expectations 36 (12)
2.6 Utility Functions 48 (2)
2.7 Integral Transforms 50 (4)
2.8 Conditional Probabilities and Expectations 54 (3)
2.9 Limit Theorems 57 (5)
2.10 Stochastic Processes 62 (6)
2.11 Recommended Literature 68 (1)
2.12 Problems 69 (8)
3 Markov Chains 77 (52)
3.1 Definitions 77 (9)
3.2 Classification of States 86 (10)
3.3 Further Examples 96 (6)
3.4 The Limiting Behaviour of Markov Chains 102(14)
3.5 Random Walks 116(6)
3.6 Recommended Literature 122(1)
3.7 Problems 123(6)
4 Markov Decision Processes 129(26)
4.1 Finite-Stage Models 130(9)
4.2 Discounted Dynamic Programming 139(7)
4.3 Further Examples 146(4)
4.4 Recommended Literature 150(1)
4.5 Problems 151(4)
5 The Exponential Distribution and Poisson 155(16)
Process
5.1 Properties of the Exponential Distribution 155(4)
5.2 The Poisson Process 159(8)
5.3 Problems 167(4)
6 Jump Markov Processes 171(24)
6.1 Definitions and Basic Results 171(6)
6.2 Inhomogeneous Processes 177(7)
6.3 Birth-and-Death Processes 184(5)
6.4 PASTA 189(1)
6.5 Recommended Literature 190(1)
6.6 Problems 190(5)
7 Elements of Queueing Theory 195(32)
7.1 Definitions and Notation 195(4)
7.2 Exponential Queueing Systems 199(15)
7.2.1 M/M/1 Systems 199(8)
7.2.2 M/M/a Systems 207(6)
7.2.3 M/M/a/N Systems 213(1)
7.3 The Machine Repair Problem 214(3)
7.4 Exponential Queueing Networks 217(5)
7.5 Recommended Literature 222(1)
7.6 Problems 223(4)
8 Elements of Renewal Theory 227(10)
8.1 Definitions and Notation. Renewal Theorems 227(8)
8.2 Problems 235(2)
9 Elements of Time Series 237(36)
9.1 Stationary Sequences 239(8)
9.2 Linear Filters and Linear Processes 247(15)
9.3 A General Approach to Time Series 262(2)
Modelling
9.4 Forecasting of Time Series 264(4)
9.5 Recommended Literature 268(1)
9.6 Problems 268(5)
10 Elements of Simulation 273(30)
10.1 Basics. Random Number Generators 273(6)
10.2 The Inverse Function Method 279(6)
10.3 The Rejection Method 285(3)
10.4 Monte Carlo. Variance Reduction Methods 288(8)
10.4.1 The Crude Monte Carlo 289(1)
10.4.2 The Stratified Sample Method 290(1)
10.4.3 The Antithetic Variables Method 291(1)
10.4.4 The Importance Sampling Method 292(4)
10.5 Markov Chain Monte Carlo 296(3)
10.6 Recommended Literature 299(1)
10.7 Problems 299(4)
11 Martingales and Stochastic Calculus 303(46)
11.1 Martingales 303(13)
11.2 The Brownian Motion Process 316(12)
11.2.1 The Main Properties of the BM Process 316(5)
11.2.2 The Path Properties 321(2)
11.2.3 The Distributions of Some RVs 323(3)
Related to the BM
11.2.4 The Three Martingales of the BM 326(2)
Process
11.3 Defining the Ito Integral 328(8)
11.4 The Ito Formula 336(4)
11.5 Stochastic Differential Equations 340(3)
11.6 Recommended Literature 343(1)
11.7 Problems 344(5)
12 Diffusion Processes 349(36)
12.1 Definitions 349(2)
12.2 Kolmogorov Differential Equations and 351(7)
Generators
12.3 Stationary Distributions 358(4)
12.4 The Method of Differential Equations 362(5)
12.5 Some Applications 367(14)
12.5.1 Branching Processes 367(4)
12.5.2 The Wright-Fisher Model 371(3)
12.5.3 The Brownian Bridge Process 374(7)
12.6 Recommended Literature 381(1)
12.7 Problems 381(4)
13 Elements of Mathematical Finance 385(48)
13.1 Introductory Remarks 385(4)
13.2 Binomial Markets 389(4)
13.3 The Single-Period Binomial Market 393(6)
13.4 Finite Single-Period Markets 399(7)
13.5 The Multi-Period Binomial Market 406(7)
13.6 Martingales and Claim Pricing 413(3)
13.7 The Black-Scholes Framework 416(9)
13.8 Pricing Barrier Options 425(4)
13.9 Recommended Literature 429(1)
13.10 Problems 430(3)
Answers to Problems 433(38)
Greek Alphabet 471(2)
Notations 473(2)
Abbreviations 475(2)
Index 477
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