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1．Review of Probability 1

1.1 Events and Probability 1

1.2 Random Variables 3

1.3 Conditional Probability and Independence 8

1.4 Solutions 10

2．Conditional Expectation 17

2.1 Conditioning on an Event 17

2.2 Conditioning on a Discrete Random Variable 19

2.3 Conditioning on an Arbitrary Random Variable 22

2.4 Conditioning on a σ-Field 27

2.5 General Properties 29

2.6 Various Exercises on Conditional Expectation 31

2.7 Solutions 33

3．Martingales in Discrete Time 45

3.1 Sequences of Random Variables 45

3.2 Filtrations 46

3.3 Martingales 48

3.4 Games of Chance 51

3.5 Stopping Times 54

3.6 Optional Stopping Theorem 58

3.7 Solutions 61

4．Martingale Inequalities and Convergence 67

4.1 Doob's Martingale Inequalities 68

4.2 Doob's Martingale Convergence Theorem 71

4.3 Uniform Integrability and L1 Convergence of Martingales 73

4.4 Solutions 80

5．Markov Chains 85

5.1 First Examples and Definitions 86

5.2 Classification of States 101

5.3 Long-Time Behaviour of Markov Chains:General Case 108

5.4 Long-Time Behaviour of Markov Chains with Finite State Space 114

5.5 Solutions 119

6．Stochastic Processes in Continuous Time 139

6.1 General Notions 139

6.2 Poisson Process 140

6.2.1 Exponential Distribution and Lack of Memory 140

6.2.2 Construction of the Poisson Process 142

6.2.3 Poisson Process Starts from Scratch at Time t 145

6.2.4 Various Exercises on the Poisson Process 148

6.3 Brownian Motion 150

6.3.1 Definition and Basic Properties 151

6.3.2 Increments of Brownian Motion 153

6.3.3 Sample Paths 156

6.3.4 Doob's Maximal L2 Inequality for Brownian Motion 159

6.3.5 Various Exercises on Brownian Motion 160

6.4 Solutions 161

7．It？ Stochastic Calculus 179

7.1 It？ Stochastic Integral:Definition 180

7.2 Examples 189

7.3 Properties of the Stochastic Integral 190

7.4 Stochastic Differential and It？ Formula 193

7.5 Stochastic Differential Equations 202

7.6 Solutions 209

Index 223