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.
 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.

efinition

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

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

Definition

(Martingale definition) A real-valued stochastic process is an ( -adapted) martingale if for any such that .

Proposition

(Maximal inequality for martingales) For a martingale and we have

Proposition

(Optional sampling theorem) Suppose that is a stopping time and Then

1. The process is a martingale.

2. If, in addition, is uniformly integrable then .

Proposition

(Convergence theorem for martingales) Let be a uniformly integrable martingale. Then

1. the limit exists a.s.

2. .

3. .

 Notation. Index. Contents.