随机分析应用导论【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Preliminaries From Calculus1

1.1 Continuous and Differentiable Functions1

1.2 Right and Left-Continuous Functions2

1.3 Variation of a Function5

1.4 Riemann Integral10

1.5 Stieltjes Integral11

1.6 Differentials and Integrals15

1.7 Taylor's Formula and other results16

2 Concepts of Probability Theory23

2.1 Discrete Probability Model23

2.2 Continuous Probability Model30

2.3 Expectation and Lebesgue Integral35

2.4 Transforms and Convergence39

2.5 Independence and Conditioning40

2.6 Stochastic Processes in Continuous Time46

3 Basic Stochastic Processes55

3.1 Brownian Motion56

3.2 Brownian Motion as a Gaussian Process58

3.3 Properties of Brownian Motion Paths61

3.4 Three Martingales of Brownian Motion63

3.5 Markov Property of Brownian Motion65

3.6 Exit Times and Hitting Times68

3.7 Maximum and Minimum of Brownian Motion70

3.8 Distribution of Hitting Times72

3.9 Reflection Principle and Joint Distributions73

3.10 Zeros of Brownian Motion.Arcsine Law74

3.11 Size of Increments of Brownian Motion77

3.12 Brownian Motion in Higher Dimensions78

3.13 Random Walk80

3.14 Stochastic Integral in Discrete Time81

3.15 Poisson Process82

3.16 Exercises85

4 Brownian Motion Calculus87

4.1 Definition of It？ Integral87

4.2 It？ integral process95

4.3 It？'s Formula for Brownian motion99

4.4 Stochastic Differentials and It？ Processes102

4.5 It？'s formula for functions of two variables109

4.6 Stochastic Exponential111

4.7 It？ Processes in Higher Dimensions112

4.8 Exercises114

5 Stochastic Differential Equations117

5.1 Definition of Stochastic Differential Equations117

5.2 Strong Solutions to SDE's120

5.3 Solutions to Linear SDE's121

5.4 Existence and Uniqueness of Strong Solutions125

5.5 Markov Property of Solutions126

5.6 Weak Solutions to SDE's128

5.7 Existence and Uniqueness of Weak Solutions130

5.8 Backward and Forward Equations134

5.9 Exercises137

6 Diffusion Processes139

6.1 Martingales and Dynkin's formula139

6.2 Calculation of Expectations and PDE's143

6.3 Homogeneous Diffusions145

6.4 Exit Times From an Interval148

6.5 Representation of Solutions of PDE's152

6.6 Explosion153

6.7 Recurrence and Transience155

6.8 Diffusion on an Interval156

6.9 Stationary Distributions157

6.10 Multidimensional SDE's160

6.11 Exercises167

7 Martingales169

7.1 Definitions169

7.2 Uniform Integrability171

7.3 Martingale Convergence173

7.4 Optional Stopping175

7.5 Localization.Local Martingales177

7.6 Quadratic Variation of Martingales180

7.7 Martingale Inequalities182

7.8 Continuous martingales184

7.9 Change of Time in SDE's185

7.10 Martingale Representations187

7.11 Exercises189

8 Calculus For Semimartingales191

8.1 Semimartingales191

8.2 Quadratic Variation and Covariation192

8.3 Predictable Processes194

8.4 Doob-Meyer Decomposition195

8.5 Definition of Stochastic Integral196

8.6 Properties of Stochastic Integrals199

8.7 It？'s Formula:continuous case200

8.8 Local Times202

8.9 Stochastic Exponential203

8.10 Compensators and Sharp Bracket Process207

8.11 It？'s Formula:general case212

8.12 Elements of the General Theory214

8.13 Exercises217

9 Pure Jump Processes219

9.1 Definitions219

9.2 Pure Jump Process Filtration220

9.3 It？'s Formula for Processes of Finite Variation221

9.4 Counting Processes222

9.5 Markov Jump Processes229

9.6 Stochastic equation for Markov Jump Processes231

9.7 Explosions in Markov Jump Processes233

9.8 Exercises234

10 Change of Probability Measure237

10.1 Change of Measure for Random Variables237

10.2 Equivalent Probability Measures238

10.3 Change of Measure for Processes240

10.4 Change of Drift in Diffusion243

10.5 Change of Wiener Measure244

10.6 Change of Measure for Point Processes245

10.7 Likelihood Ratios247

10.8 Exercises250

11 Applications in Finance253

11.1 Financial Derivatives and Arbitrage253

11.2 A Finite Market Model256

11.3 Semimartingale Market Model260

11.4 Diffusion and Black-Scholes Model265

11.5 Interest Rates Models273

11.6 Options,Caps,Floors,Swaps and Swaptions281

11.7 Exercises283

12 Applications in Biology289

12.1 Branching Diffusion289

12.2 Wright-Fisher Diffusion292

12.3 Birth-Death Processes293

12.4 Exercises297

13 Applications in Engineering and Physics299

13.1 Filtering299

13.2 Stratanovich Calculus304

13.3 Random Oscillators305

13.4 Exercises313

References315