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.

## Feller process and semi-group resolvent.

efinition

(Feller process) Let be a Markov process in with a homogeneous transition function . The is called "Feller process" if the associated backward propagator has the following property:

Definition

(Resolvent of Feller process) For a Feller process in we define the "resolvent" :

Proposition

(Properties of Feller resolvent 1)

1. , , , .

2. .

3. The range does not depend on .

Proof

Let . We verify (2) as follows: The last expression is symmetrical with respect to and and permits the change of integration order (see the proposition ( Fubini theorem )).

To prove (1) we calculate We change the order of integration: We make a change of variables in the internal integral: We use the property and linearity of :

(3) is the consequence of (1).

Proposition

(Feller process property 1) Let be a stochastic process in with a homogeneous propagator . is a Feller process iff

1. .

2. .

Proof

For we calculate We make the change in the first integral.

It follows from (1) that is a linear operator in and we conclude from the definition of that

Hence, for Therefore thus In view of the proposition ( Properties of Feller resolvent 1 )-3, it remains to prove that is dense in . We evaluate for large and aim to show that the quantity is small. We make a change in the integral, , The quantity is positive and tends to 0 when tends to . Hence, using (2), In addition Therefore, by the proposition ( Dominated convergence theorem ), for any finite measure Hence, if the measure vanishes on then it vanishes on . Thus, is dense in .

Proposition

(Properties of Feller resolvent 2) Let be a resolvent of a Feller process. We have

Proof

According to the proof of the proposition ( Feller process property 1 ), We continue Hence and the statement follows by the definition ( Feller process ) and the proposition ( Dominated convergence theorem ).

 Notation. Index. Contents.