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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.

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.


(Backward Kolmogorov for Feller process) Let $X_{t}$ be a Feller process in $\QTR{cal}{R}^{n}$ , $L$ is the associated backward generator and MATH . We have

0. MATH .

1. MATH , $\forall t>0$ .

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


4. The set $D\left( L\right) $ is dense in MATH .

5. $L$ is a closed operator in MATH .


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

(2) MATH

(3) MATH

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

(5). We introduce the notations MATH Let $f_{n}\rightarrow f$ , MATH and MATH . We aim to show that MATH and $Lf=g$ . We calculate MATH since MATH MATH Note that MATH hence MATH and we conclude MATH and $Lf=g$ .


(Properties of Feller resolvent 3) For every $p>0$ the map $f\rightarrow pf-Lf$ from $D\left( L\right) $ to MATH is one-to-one and onto and its inverse is $R_{p}$ . (see the definition ( Resolvent of Feller process ))


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

Both properties are direct verification.


(Dissipation property of Feller process) Let $L$ be a generator of a Feller process in $\QTR{cal}{R}^{n}$ , MATH and MATH is such that MATH . Then MATH


By definition MATH and MATH


(Martingale property of Feller process) Let $L$ be a generator of the Feller process $X_{t}$ in $\QTR{cal}{R}^{n}$ , $X_{0}=x$ , MATH and $\QTR{cal}{F}_{t}$ is the $X_{t}$ -generated filtration. Then the process MATH is an $\QTR{cal}{F}_{t}$ -martingale.

Conversely, if MATH and there exists a MATH such that MATH is an $\QTR{cal}{F}_{t}$ -martingale for every $x=X_{0}$ then MATH and $Lf=g$ .


To prove the direct statement we calculate for $0<t<T$ : MATH By definition of $P_{t}$ and Markovian property of $X_{t}$ , MATH thus MATH It remains to note that according to the proposition ( Backward Kolmogorov for Feller process )-3, MATH Hence, we make a change $s-t=\tau$ in the integral: MATH and conclude MATH To prove the converse statement we rewrite the condition MATH as MATH Hence, we verify the conclusion of the converse statement as follows MATH Hence, MATH and $Lf=g$ .

Notation. Index. Contents.

Copyright 2007