Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 be a Markov process in and are the associated propagators. We define the backward and forward "generators" and as follows:

1. , (Backward Kolmogorov generator) .

2. , (Forward Kolmogorov generator) .

A function is included in the domain of if the limit 1 exists.

A measure is included in the domain of if the limit 2 exists.

Proposition

(Kolmogorov equations in general setting). Let be a Markov process in and is the associated transition function. For any , and we have

Claim

1. (Generic backward Kolmogorov equation) and

 (Generic Backward Kolmogorov)

2.(Generic forward Kolmogorov equation) and

 (Generic Forward Kolmogorov)

Proof

1.

2.

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