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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
1. Conditional probability.
2. Normal distribution.
3. Brownian motion.
A. Definition of standard Brownian motion.
B. Brownian motion passing through gates.
C. Reflection principle.
D. Brownian motion hitting a barrier.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Definition of standard Brownian motion.

et $\Omega$ be a random event space (see the section ( Filtration )) and $\xi$ be a standard normal variable (see the section ( Normal variable )) . The standard Brownian motion $W_{t}$ is a mapping MATH with the following properties:

  1. The path MATH is continuous for every $\omega\in\Omega$ .

  2. The random variable $W_{t}-W_{s}$ is independent from MATH (see the section ( Filtration )) for every $t,s$ , $0<s<t$ .

  3. For any $s,t:t>s>0$ , the random variable $W_{t}-W_{s}$ is distributed as $\sqrt{t-s}\xi$ :

    MATH (Brownian motion)

  4. MATH

The argument $t$ of the Brownian motion $W_{t}$ is conventionally called "time".

Notation. Index. Contents.

Copyright 2007