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.
 9 Levy process.
 A. Infinitely divisible distributions and Levy-Khintchine formula.
 B. Generator of Levy process.
 C. Poisson point process.
 D. Construction of generic Levy process.
 E. Subordinators.
 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.

## Construction of generic Levy process.

roposition

(Construction of generic Levy process) Let , and is a measure on such that . Then there exists a filtered probability space and a Levy process such that

Proof

Let be a standard Brownian motion in and is an independent from Poisson point process in (see the definition ( Poisson point process )) with characteristic measure and . We introduce the processes The family has an limit with respect to the norm for any . The convergence in insures that the limit is a Levy process.

The processes have the characteristic exponents : Thus is a sum of the independent processes and has the characteristic exponent as claimed.

We calculate the characteristic functions as follows. To calculate we note the definition ( Characteristic measure of Poisson point process ): Hence, for small we use the definition ( Poisson point process )-2,5: Thus The above means that the probability that twice within a small time interval is negligible. Similarly, For the point (see the definition ( Poisson point process )) we have The rest of the calculation follows the technique developed in the chapter ( Poisson process ). For a fixed we set , , Note that thus and we continue Note that the last expression is of the form we perform the Taylor expansion of log: Hence where the term disapeared because can be of any value and we let .

The expectation is evaluated similarly.

 Notation. Index. Contents.