1 Preliminaries From Calculus

1.1 Continuous and Differentiable Functions

1.2 Right and Left-Continuous Functions

1.3 Variation of a Function

1.4 Riemann Integral

1.5 Stieltjes Integral

1.6 Differentials and Integrals

1.7 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 Conditioning

2.6 Stochastic Processes in Continuous Time

3 Basic Stochastic Processes

3.1 Brownian Motion

3.2 Brownian Motion as a Gaussian Process

3.3 Properties of Brownian Motion Paths

3.4 Three Martingales of Brownian Motion

3.5 Markov Property of Brownian Motion

3.6 Exit Times and Hitting Times

3.7 Maximum and Minimum of Brownian Motion

3.8 Distribution of Hitting Times

3.9 Reflection Principle and Joint Distributions

3.10 Zeros of Brownian Motion.Arcsine Law

3.11 Size of Increments of Brownian Motion

3.12 Brownian Motion in Higher Dimensions

3.13 Random Walk

3.14 Stochastic Integral in Discrete Time

3.15 Poisson Process

3.16 Exercises

4 Brownian Motion Calculus

4.1 Definition of It？ Integral

4.2 It？ integral process

4.3 It？'s Formula for Brownian motion

4.4 Stochastic Differentials and It？ Processes

4.5 It？'s formula for functions of two variables

4.6 Stochastic Exponential

4.7 It？ Processes in Higher Dimensions

4.8 Exercises

5 Stochastic Differential Equations

5.1 Definition of Stochastic Differential Equations

5.2 Strong Solutions to SDE's

5.3 Solutions to Linear SDE's

5.4 Existence and Uniqueness of Strong Solutions

5.5 Markov Property of Solutions

5.6 Weak Solutions to SDE's

5.7 Existence and Uniqueness of Weak Solutions

5.8 Backward and Forward Equations

5.9 Exercises

6 Diffusion Processes

6.1 Martingales and Dynkin's formula

6.2 Calculation of Expectations and PDE's

6.3 Homogeneous Diffusions

6.4 Exit Times From an Interval

6.5 Representation of Solutions of PDE's

6.6 Explosion

6.7 Recurrence and Transience

6.8 Diffusion on an Interval

6.9 Stationary Distributions

6.10 Multidimensional SDE's

6.11 Exercises

7 Martingales

7.1 Definitions

7.2 Uniform Integrability

7.3 Martingale Convergence

7.4 Optional Stopping

7.5 Localization.Local Martingales

7.6 Quadratic Variation of Martingales

7.7 Martingale Inequalities

7.8 Continuous martingales

7.9 Change of Time in SDE's

7.10 Martingale Representations

7.11 Exercises

8 Calculus For Semimartingales

8.1 Semimartingales

8.2 Quadratic Variation and Covariation

8.3 Predictable Processes

8.4 Doob-Meyer Decomposition

8.5 Definition of Stochastic Integral

8.6 Properties of Stochastic Integrals

8.7 It？'s Formula:continuous case

8.8 Local Times

8.9 Stochastic Exponential

8.10 Compensators and Sharp Bracket Process

8.11 It？'s Formula:general case

8.12 Elements of the General Theory

8.13 Exercises

9 Pure Jump Processes

9.1 Definitions

9.2 Pure Jump Process Filtration

9.3 It？'s Formula for Processes of Finite Variation

9.4 Counting Processes

9.5 Markov Jump Processes

9.6 Stochastic equation for Markov 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 Equivalent Probability Measures

10.3 Change of Measure for Processes

10.4 Change of Drift in Diffusion

10.5 Change of Wiener Measure

10.6 Change of Measure for Point Processes

10.7 Likelihood Ratios

10.8 Exercises

11 Applications in Finance

11.1 Financial Derivatives and Arbitrage

11.2 A Finite Market Model

11.3 Semimartingale Market Model

11.4 Diffusion and Black-Scholes Model

11.5 Interest Rates Models

11.6 Options,Caps,Floors,Swaps and Swaptions

11.7 Exercises

12 Applications in Biology

12.1 Branching Diffusion

12.2 Wright-Fisher Diffusion

12.3 Birth-Death Processes

12.4 Exercises

13 Applications in Engineering and Physics

13.1 Filtering

13.2 Stratanovich Calculus

13.3 Random Oscillators

13.4 Exercises

References