1 Preliminaries From Calculus
1.1 Functions in Calculus
1.2 Variation of a Function
1.3 Riemann Integral and Stieltjes Integral
1.4 Lebesgue’s Method of Integration
1.5 Differentials and Integrals
1.6 Taylor’s Formula and Other Results
2 Concepts of Probability Theory
2.1 Discrete Probability Model
2.2 Continuous Probability Model
2.3 Expectation and Lebesgue Integral
2.4 Transforms and Convergence
2.5 Independence and Covariance
2.6 Normal （Gaussian） Distributions
2.7 Conditional Expectation
2.8 Stochastic Processes in Continuous Time
3 Basic Stochastic Processes
3.1 Brownian Motion
3.2 Properties of Brownian Motion Paths
3.3 Three Martingales of Brownian Motion
3.4 Markov Property of Brownian Motion
3.5 Hitting Times and Exit Times
3.6 Maximum and Minimum of Brownian Motion
3.7 Distribution of Hitting Times
3.8 Reflection Principle and Joint Distributions
3.9 Zeros of Brownian Motion. Arcsine Law
3.10 Size of Increments of Brownian Motion
3.11 Brownian Motion in Higher Dimensions
3.12 Random Walk
3.13 Stochastic Integral in Discrete Time
3.14 Poisson Process
3.15 Exercises
4 Brownian Motion Calculus
4.1 Definition of It6 Integral
4.2 Ito Integral Process
4.3 Ito Integral and Gaussian Processes
4.4 Ito’s Formula for Brownian Motion
4.5 Ito Processes and Stochastic Differentials
4.6 Ito’s Formula for It6 Processes
4.7 Ito Processes in Higher Dimensions
4.8 Exercises
5 Stochastic Differential Equations
5.1 Definition of Stochastic Differential Equations
5.2 Stochastic Exponential and Logarithm
5.3 Solutions to Linear SDEs
5.4 Existence and Uniqueness of Strong Solutions
5.5 Markov Property of Solutions
5.6 Weak Solutions to SDEs
5.7 Construction of Weak Solutions
5.8 Backward and Forward Equations
5.9 Stratanovich Stochastic Calculus
5.10 Exercises
6 Diffusion Processes
6.1 Martingales and Dynkin’s Formula
6.2 Calculation of Expectations and PDEs
6.3 Time Homogeneous Diffusions
6.4 Exit Times from an Interval
6.5 Representation of Solutions of ODEs
6.6 Explosion
6.7 Recurrence and Transience
6.8 Diffusion on an Interval
6.9 Stationary Distributions
6.10 Multi-Dimensional SDEs
6.11 Exercises
7 Martingales
7.1 Definitions
7.2 Uniform Integrability
7.3 Martingale Convergence
7.4 Optional Stopping
7.5 Localization and Local Martingales
7.6 Quadratic Variation of Martingales
7.7 Martingale Inequalities
7.8 Continuous Martingales. Change of Time
7.9 Exercises
8 Calculus For Semimartingales
8.1 Semimartingales
8.2 Predictable Processes
8.3 Doob-Meyer Decomposition
8.4 Integrals with respect to Semimartingales
8.5 Quadratic Variation and Covariation
8.6 ItS’s Formula for Continuous Semimartingales
8.7 Local Times
8.8 Stochastic Exponential
8.9 Compensators and Sharp Bracket Process
8.10 ItS’s Formula for Semimartingales
8.11 Stochastic Exponential and Logarithm
8.12 Martingale （Predictable） Representations
8.13 Elements of the General Theory
8.14 Random Measures and Canonical Decomposition
8.15 Exercises
9 Pure Jump Processes
9.1 Definitions
9.2 Pure Jump Process Filtration
9.3 ItS’s Formula for Processes of Finite Variation
9.4 Counting Processes
9.5 Markov Jump Processes
9.6 Stochastic Equation for Jump Processes
9.7 Explosions in Markov Jump Processes
9.8 Exercises
10 Change of Probability Measure
10.1 Change of Measure for Random Variables
10.2 Change of Measure on a General Space
10.3 Change of Measure for Processes
10.4 Change of Wiener Measure
10.5 Change of Measure for Point Processes
10.6 Likelihood Functions
10.7 Exercises
11 Applications in Finance： Stock and FX Options
11.1 Financial Deriwtives and Arbitrage
11.2 A Finite Market Model
11.3 Semimartingale Market Model
11.4 Diffusion and the Black-Scholes Model
11.5 Change of Numeraire
11.6 Currency （FX） Options
11.7 Asian, Lookback and Barrier Options
11.8 Exercises
12 Applications in Finance： Bonds, Rates and Option
12.1 Bonds and the Yield Curve
12.2 Models Adapted to Brownian Motion
12.3 Models Based on the Spot Rate
12.4 Merton’s Model and Vasicek’s Model
12.5 Heath-Jarrow-Morton （HJM） Model
12.6 Forward Measures. Bond as a Numeraire
12.7 Options, Caps and Floors
12.8 Brace-Gatarek-Musiela （BGM） Model
12.9 Swaps and Swaptions
12.10 Exercises
13 Applications in Biology
13.1 Feller’s Branching Diffusion
13.2 Wright-Fisher Diffusion
13.3 Birth-Death Processes
13.4 Branching Processes
13.5 Stochastic Lotka-Volterra Model
13.6 Exercises
14 Applications in Engineering and Physics
14.1 Filtering
14.2 Random Oscillators
14.3 Exercises
Solutions to Selected Exercises
References
Index
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