Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Martingales and stopping times.


(Stochastic process) Let MATH be a probability space and MATH be a measurable space. The family MATH of measurable mappings MATH is called "stochastic process". The pair MATH is called "state space". The $t$ is the time parameter. For a fixed $\omega\in\Omega$ the mapping MATH is called "path" of the process.

The notions of filtration and stopping time where introduced in the section ( Filtration_definition_section ).


(Martingale definition) A real-valued stochastic process $M_{t}$ is an ( MATH -adapted) martingale if MATH for any $s,t$ such that $0\leq s\leq t$ .


(Maximal inequality for martingales) For a martingale $M_{t}$ and $t>0$ we have MATH


(Optional sampling theorem) Suppose that $\theta$ is a stopping time and MATH Then

1. The process $M_{\theta\wedge t}$ is a martingale.

2. If, in addition, $M_{t}$ is uniformly integrable then MATH .


(Convergence theorem for martingales) Let $M_{t}$ be a uniformly integrable martingale. Then

1. the limit MATH exists a.s.

2. MATH .

3. MATH .

Notation. Index. Contents.

Copyright 2007