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.
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.

Conditional probability II.


he conditional probability was introduced on intuitive level in the chapter ( Conditional probability chapter ).

Definition

(Conditional expectation) Let MATH be the probability space, $\QTR{cal}{G}$ be a $\sigma$ -subalgebra of $\QTR{cal}{F}$ and $X$ is a r.v. The conditional expectation MATH is a r.v. that satisfies the following two conditions:

1. MATH is $\QTR{cal}{G}$ -measurable.

2. MATH .

Proposition

(Existence of conditional expectation) If MATH and $\QTR{cal}{G}$ is a $\sigma$ -subalgebra of $\QTR{cal}{F}$ then there exists MATH . The MATH is unique almost surely.

Proof

The mapping MATH given by MATH satisfies conditions of the proposition ( Radon-Nikodym theorem ) applied to $\mu$ and $P$ on $\QTR{cal}{G}$ . Hence, there exists a $\QTR{cal}{G}$ -measurable function $d\mu/dP$ such that MATH The function $d\mu/dP$ satisfies the definition ( Conditional expectation ).

Example

Let $Y$ be a discrete valued r.v. MATH Then the sets MATH generate a $\sigma$ -algebra $\QTR{cal}{G}$ . Let $X$ be a r.v. MATH . The r.v. MATH satisfies conditions of the definition ( Conditional expectation ).

Notation

Let MATH be the $\sigma$ -algebra generated by the r.v. $X$ . We denote MATH

Proposition

(Mapping of r.v.) If MATH then MATH for some extended valued Borel measurable function $\phi$ .

Proof

If $Z$ is not bounded then we consider the sequence MATH and thus restrict to bounded r.v. If $Z$ has a variable sign then we consider $Z^{+}$ and $Z^{-}$ . Thus, it is enough to prove the proposition for a bounded positive variable.

Let MATH be a sequence of r.v. s.t. $Z_{m}\uparrow Z$ everywhere and MATH for MATH and MATH . Then MATH MATH and MATH . We set MATH

Proposition

(Mapping of conditional expectation) For a MATH there exists a Borel measurable function $\phi$ such that MATH For measures $\lambda,\mu$ defined by MATH we have MATH

Proof

The existence of $\phi$ follows from the proposition ( Mapping of r.v. ).

The second claim is equivalent to the claim MATH Thus, it follows from the definition ( Conditional expectation ) and the proposition ( Radon-Nikodym theorem ).

Proposition

(Basic properties of conditional expectation) Let MATH and MATH and $X$ is $\QTR{cal}{G}$ -measurable. Then almost surely we have:

1. MATH .

2. MATH .

3. MATH is a linear operation.

4. MATH .

5. MATH .

6. MATH .

7. MATH .

8. If $\phi$ is a convex function and MATH then MATH

Proof

Direct verification from the definition accompanied by uniqueness of conditional expectation almost surely. The (7) follows from the proposition ( Dominated convergence theorem ). The (8) follows by the proposition ( Jensen inequality ).





Notation. Index. Contents.


















Copyright 2007