Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
I. Basic math.
1. Conditional probability.
2. Normal distribution.
A. Definition of normal variable.
B. Linear transformation of random variables.
C. Multivariate normal distribution. Choleski decomposition.
D. Calculus of normal variables.
E. Central limit theorem (CLT).
3. Brownian motion.
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.

Linear transformation of random variables.


e introduce the notation MATH to describe the statement "random variable $Y$ has distribution $g\left( y\right) $ ".

Proposition

Let $X$ be a vector of random variables and MATH for some function MATH . Set $Y=AX+b$ for some deterministic square matrix $A$ and vector $b$ . If $\det A\neq0$ then MATH

Proof

For any domain $D$ of the $y$ -space we can write MATH MATH We make the change of variables $y=Ax+b$ in the last integral.

MATH (Linear transformation of random variables)
Hence MATH

The linear transformation $\sigma\xi+\mu$ is distributed as MATH . The $\xi$ was defined in the section ( Definition of normal variable ).

For two independent standard normal variables (s.n.v.) $\xi_{1}$ and $\xi _{2},$ the combination MATH is distributed as MATH .

A product of normal variables is not a normal variable. See the section ( Chi squared distribution ).





Notation. Index. Contents.


















Copyright 2007