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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 $X_{t}$ be a Markov process in $\QTR{cal}{R}^{n}$ with a homogeneous transition function $\pi$ . The $X_{t}$ is called "Feller process" if the associated backward propagator $P_{t}$ has the following property: MATH

Definition

(Resolvent of Feller process) For a Feller process $X_{t}$ in $\QTR{cal}{R}^{n}$ we define the "resolvent" MATH : MATH

Proposition

(Properties of Feller resolvent 1)

1. MATH , $\forall q>0$ , $\forall p>0$ , $q\not =p$ .

2. MATH .

3. The range MATH does not depend on $p$ .

Proof

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

To prove (1) we calculate MATH We change the order of integration: MATH We make a change of variables $t-s=\tau$ in the internal integral: MATH We use the property MATH and linearity of $P$ : MATH

(3) is the consequence of (1).

Proposition

(Feller process property 1) Let $X_{t}$ be a stochastic process in $\QTR{cal}{R}^{n}$ with a homogeneous propagator $P_{t}$ . $X_{t}$ is a Feller process iff

1. MATH .

2. MATH MATH .

Proof

For MATH we calculate MATH We make the change $y=s+t$ in the first integral. MATH

It follows from (1) that $P_{t}$ is a linear operator in MATH and we conclude from the definition of $P_{t}$ that MATH

Hence, for MATH MATH Therefore MATH thus MATH In view of the proposition ( Properties of Feller resolvent 1 )-3, it remains to prove that MATH is dense in MATH . We evaluate MATH for large $p$ and aim to show that the quantity is small. We make a change $e^{-ps}=z$ in the integral, $-pe^{-ps}ds=dz$ , MATH MATH The quantity $-\frac{1}{p}\ln z$ is positive and tends to 0 when $p$ tends to $+\infty$ . Hence, using (2), MATH In addition MATH Therefore, by the proposition ( Dominated convergence theorem ), for any finite measure $\mu$ MATH Hence, if the measure $\mu$ vanishes on MATH then it vanishes on MATH . Thus, MATH is dense in MATH .

Proposition

(Properties of Feller resolvent 2) Let $U_{p}$ be a resolvent of a Feller process. We have MATH

Proof

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





Notation. Index. Contents.


















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