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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
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.



(Subordination) Let $X_{t}$ be a Markov process, $X_{0}=0$ and $T_{t}$ is a stochastic process MATH . In addition, we assume that $T$ and $X$ are independent. We define the "subordinated" process $X_{T_{t}}$ .

The process $X_{T_{t}}$ does not need to be Markovian.

We proceed to investigate conditions for $X_{T_{t}}$ to be Markovian.

Let MATH be the transition function for the process $X$ and MATH be the transition function for the process $T$ . We calculate the transition function for the subordinated process $X_{T_{t}}$ . For $0<s<t$ we define a uniform mesh over $[0,+\infty)$ with step $h,~t_{k+1}-t_{k}=h$ and apply the formula ( Total_probability_rule ): MATH By independence of $X_{t}$ and $T_{t}$ we simplify MATH We need the expression Prob MATH to be well defined for any $t,s:0<s<t$ and any however fine mesh MATH . This means that the law of $T_{t}-T_{s}$ needs to be independent of MATH . Thus MATH needs to be a function of the form MATH We continue MATH Next, we would like to write MATH This imposes regularity requirements MATH along $t$ -parameter. We pass to the limit $h\rightarrow0$ : MATH For $X_{T_{t}}$ to be Markovian it needs to satisfy the formula ( Kolmogorov-Chapman equation ), $0<s<u<t$ : MATH We calculate the integral MATH Note that by assumption, $X_{t}$ is Markovian and has the formula ( Kolmogorov-Chapman equation ): MATH Hence, we continue MATH We make a change of variables MATH MATH and change the order of integration MATH For $x_{T_{t}}$ to be Markovian the last expression should be equal to MATH thus we need MATH The convolution on the right is a formula for distribution of a sum of two independent r.v. We conclude that the increments of $T_{t}$ should be infinitely divisible and stationary. We summarize these findings in the following proposition.


Let $X_{t}$ be a Markov process with $t$ -differentiable transition function MATH and $T_{t}$ is a Levy process with non-negative increments. Then the subordinated process $X_{T_{t}}$ is Markovian.

Notation. Index. Contents.

Copyright 2007