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Martingales, Diffusions and Financial Mathematics

Notes on stochastic integration relative to semimartingales

There are many books on financial calculus. Some of them are written from the perspective of differential equations. The books mentioned are of course written from a probabilistic point of view.

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Contents

1. Measure Theory

1.1. Conditional Expectation

1.2. Uniform Integrability

1.3. Monotone Class Theorem

2. Discrete Time Martingales

2.1. Martingales

2.2. Stopped Martingales

2.3. Martingale Transforms

2.4. Doob’s Upcrossing Inequality

2.5. Martingale Convergence

2.6. Reverse Martingale Convergence

2.7. Doob Decomposition

2.8. Optional Stopping

2.9. Maximal Inequalities

3. Discrete Time Option Pricing

4. Continuous Time Martingales

4.1. Stochastic Processes

4.2. Martingales

4.3. Martingale Convergence

4.4. Stopping

4.5. Brownian Motion

4.6. Local Martingales

4.7. Maximal Inequalities

5. Stochastic Integrals

5.1. Predictable Sets and Processes

5.2. Dol´eans Measure

5.3. Square-integrable Martingales

5.4. Locally Square-integrable Martingales

5.5. Brownian Motion

5.6. Martingales of Bounded Variation

5.7. Semimartingales

5.8. Quadratic Variation

5.9. Predictable Quadratic Variation

5.10. Itˆo’s Formula for Continuous Processes

5.11. Space of Square-integrable Martingales

5.12. Itˆo’s Formula

6. Stochastic Calculus

6.1. L´evy’s Theorem

6.2. Brownian Martingales

6.3. Exponential Processes

6.4. Cameron-Martin-Girsanov Theorem

7. Stochastic Differential Equations

7.1. Strong Solutions

7.2. Martingale Problem and Weak Solutions

7.3. Markov Property

8. Option Pricing in Continuous Time

9. Random Measures

9.1. Compensators

9.2. Marked Point Processes

9.3. Jump Measure

9.4. Change of Measure

9.5. Reduction of Flow

9.6. Stochastic Integrals

10. Stochastic Calculus

10.1. Characteristics

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