Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 A. Operations on sets and logical statements.
 B. Fundamental inequalities.
 C. Function spaces.
 D. Measure theory.
 E. Various types of convergence.
 F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
 G. Lebesgue differentiation theorem.
 H. Fubini theorem.
 I. Arzela-Ascoli compactness theorem.
 J. Partial ordering and maximal principle.
 K. Taylor decomposition.
 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.

## Partial ordering and maximal principle.

efinition

The relation is said to be a partial ordering on the set A iff it has the following properties

1.

2.

The partial ordering is said to be a linear ordering iff either or for any two elements of the set A.

Any linearly ordered subset C of a partially ordered set A is called a chain. The chain C is said to be maximal if it is not a nontrivial subset of any other chain.

For a chain C we define the upper bound as an element with the property for any .

The element of the partially ordered set A is called maximal iff

Axiom

(Axiom of choice). Let be an arbitrary index set . Suppose that for any a set is given. Then there exists a map acting on such that .

Lemma

(Hausdorff maximal principle). Every chain of a partially ordered set is contained in some maximal chain.

Lemma

(Zorn maximal principle). If every chain of a partially ordered set has a upper bound then there exists a maximal element.

Both versions of the maximal principle are consequences of the axiom of choice.

 Notation. Index. Contents.