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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.

## Example: backward Kolmogorov generator for Ito process with jump. e use technique of the previous section to calculate the operator ( Backward Kolmogorov generator ) for the process where the Poisson process has the intensity and is conditionally independent from .

Similarly to the previous section, In order to calculate the above expectation we use the recipe ( Total_probability_rule ). We consider the disjoint events the process does not jump during , the process jumps once during and . . According to the chapter ( Poisson process ), By the conditional independence of and , the diffusion terms in are small compared to the jump and do not survive the procedure . Hence, In addition, Consequently, The above results agrees with the section ( Backward_equation_for_jump_diffusion ).

 Notation. Index. Contents.
 Copyright 2007