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.
A. Definition of conditional probability.
B. A bomb on a plane.
C. Dealing a pair in the "hold' em" poker.
D. Monty-Hall problem.
E. Two headed coin drawn from a bin of fair coins.
F. Randomly unfair coin.
G. Recursive Bayesian calculation.
H. Birthday problem.
I. Backward induction.
J. Conditional expectation. Filtration. Flow of information. Stopping time.
2. Normal distribution.
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.

Definition of conditional probability.


et $A$ and $B$ be two events of the same event space MATH , where $\Omega$ is the total event space, $\Sigma$ is a collection of subsets of $\Omega$ equipped with set operations and $P$ is a set function acting MATH with the properties MATH for any at most countable collection of sets MATH such that MATH MATH .

Conditional probability $P(A|B)$ of $A$ conditionally on $B$ is defined as

MATH (Bayes formula)

If the whole space $\Omega$ is represented as some disjoint union MATH then for any $A$

MATH (Total probability rule)
The last relationship is called "the total probability rule".

Conditional probability has transitive property: MATH hence,

MATH (Transitive bayes formula)

Suppose we are facing calculation of the quantity MATH and the quantity MATH is easily computable. The repeated application of the Bayes formula expresses MATH in terms of MATH . Indeed, the Bayes formula is symmetrical: MATH hence

MATH (Inversion remark)

The properties ( Total_probability_rule ),( Transitive_Bayes_formula ) and ( Inversion_remark ) are used repeatedly in the following chapters.





Notation. Index. Contents.


















Copyright 2007