Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
a. Example: backward Kolmogorov generator for diffusion.
b. Example: backward Kolmogorov generator for Ito process with jump.
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.


efinition

(Forward and backward generators) Let $X_{t}$ be a Markov process in $\QTR{cal}{R}^{n}$ and $P_{s,t}^{\ast},$ $P_{s,t}$ are the associated propagators. We define the backward and forward "generators" $L_{t}$ and $L_{t}^{\ast}$ as follows:

1. MATH , (Backward Kolmogorov generator) .

2. MATH , (Forward Kolmogorov generator) .

A function $f$ is included in the domain MATH of $L_{t}$ if the limit 1 exists.

A measure $m$ is included in the domain MATH of $L_{t}^{\ast}$ if the limit 2 exists.

Proposition

(Kolmogorov equations in general setting). Let $X_{t}$ be a Markov process in $\QTR{cal}{R}^{n}$ and $\pi$ is the associated transition function. For any MATH , MATH and $0<s<t$ we have

Claim

1. (Generic backward Kolmogorov equation) MATH and

MATH (Generic Backward Kolmogorov)

2.(Generic forward Kolmogorov equation) MATH and

MATH (Generic Forward Kolmogorov)

Proof

1. MATH

2. MATH




a. Example: backward Kolmogorov generator for diffusion.
b. Example: backward Kolmogorov generator for Ito process with jump.

Notation. Index. Contents.


















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