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 be the probability space, be a -subalgebra of and is a r.v. The conditional expectation is a r.v. that satisfies the following two conditions:

1. is -measurable.

2. .

Proposition

(Existence of conditional expectation) If and is a -subalgebra of then there exists . The is unique almost surely.

Proof

The mapping given by satisfies conditions of the proposition ( Radon-Nikodym theorem ) applied to and on . Hence, there exists a -measurable function such that The function satisfies the definition ( Conditional expectation ).

Example

Let be a discrete valued r.v. Then the sets generate a -algebra . Let be a r.v. . The r.v. satisfies conditions of the definition ( Conditional expectation ).

Notation

Let be the -algebra generated by the r.v. . We denote

Proposition

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

Proof

If is not bounded then we consider the sequence and thus restrict to bounded r.v. If has a variable sign then we consider and . Thus, it is enough to prove the proposition for a bounded positive variable.

Let be a sequence of r.v. s.t. everywhere and for and . Then and . We set

Proposition

(Mapping of conditional expectation) For a there exists a Borel measurable function such that For measures defined by we have

Proof

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

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

Proposition

(Basic properties of conditional expectation) Let and and is -measurable. Then almost surely we have:

1. .

2. .

3. is a linear operation.

4. .

5. .

6. .

7. .

8. If is a convex function and then

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.