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.
 A. Forward and backward propagators.
 B. Feller process and semi-group resolvent.
 C. Forward and backward generators.
 D. Forward and backward generators for Feller 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.

## Forward and backward generators for Feller process.

he notion of generator was introduced in the section ( Forward and backward generators ). If the process has a homogeneous transition function then the generator does not depend on the time parameter.

Proposition

(Backward Kolmogorov for Feller process) Let be a Feller process in , is the associated backward generator and . We have

0. .

1. , .

2. The function is differentiable with respect to the strong topology in and

3.

4. The set is dense in .

5. is a closed operator in .

Proof

(0),(1) and the limit exists because :

(2)

(3)

The claim (4) follows from the two observations: and

(5). We introduce the notations Let , and . We aim to show that and . We calculate since Note that hence and we conclude and .

Proposition

(Properties of Feller resolvent 3) For every the map from to is one-to-one and onto and its inverse is . (see the definition ( Resolvent of Feller process ))

Proof

Given the proposition ( Properties of Feller resolvent 2 ), it suffices to show that we have and for we have .

Both properties are direct verification.

Proposition

(Dissipation property of Feller process) Let be a generator of a Feller process in , and is such that . Then

Proof

By definition and

Proposition

(Martingale property of Feller process) Let be a generator of the Feller process in , , and is the -generated filtration. Then the process is an -martingale.

Conversely, if and there exists a such that is an -martingale for every then and .

Proof

To prove the direct statement we calculate for : By definition of and Markovian property of , thus It remains to note that according to the proposition ( Backward Kolmogorov for Feller process )-3, Hence, we make a change in the integral: and conclude To prove the converse statement we rewrite the condition as Hence, we verify the conclusion of the converse statement as follows Hence, and .

 Notation. Index. Contents.