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.

## Subordinators.

efinition

(Subordination) Let be a Markov process, and is a stochastic process . In addition, we assume that and are independent. We define the "subordinated" process .

The process does not need to be Markovian.

We proceed to investigate conditions for to be Markovian.

Let be the transition function for the process and be the transition function for the process . We calculate the transition function for the subordinated process . For we define a uniform mesh over with step and apply the formula ( Total_probability_rule ): By independence of and we simplify We need the expression Prob to be well defined for any and any however fine mesh . This means that the law of needs to be independent of . Thus needs to be a function of the form We continue Next, we would like to write This imposes regularity requirements along -parameter. We pass to the limit : For to be Markovian it needs to satisfy the formula ( Kolmogorov-Chapman equation ), : We calculate the integral Note that by assumption, is Markovian and has the formula ( Kolmogorov-Chapman equation ): Hence, we continue We make a change of variables and change the order of integration For to be Markovian the last expression should be equal to thus we need The convolution on the right is a formula for distribution of a sum of two independent r.v. We conclude that the increments of should be infinitely divisible and stationary. We summarize these findings in the following proposition.

Proposition

Let be a Markov process with -differentiable transition function and is a Levy process with non-negative increments. Then the subordinated process is Markovian.

 Notation. Index. Contents.