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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
1. Conditional probability.
2. Normal distribution.
3. Brownian motion.
4. Poisson process.
A. Definition of Poisson process.
B. Distribution of Poisson process.
C. Poisson stopping time.
D. Arrival of k-th Poisson jump. Gamma distribution.
E. Cox process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Arrival of k-th Poisson jump. Gamma distribution.

he purpose of this section is to calculate distribution of time MATH of exactly $k$ -th jump of Poisson process.

Note that the Prob MATH is not equal to Prob MATH . Indeed, the MATH means that k or more jumps occurred before $T$ . The event MATH means that exactly k jumps occurred before $T$ . MATH and the union is disjoint. Hence, MATH We use the result ( Poisson property 3 ): MATH Therefore MATH Hence, the distribution density of the k-th arrival time is MATH Such distribution is called the "Gamma distribution". We will be using the following notation MATH Note that by normalization we must have MATH hence MATH The integral is called "Gamma function" with the traditional notation MATH This above expression expands factorial to real and complex numbers.

Notation. Index. Contents.

Copyright 2007