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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 of exactly -th jump of Poisson process.

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

 Notation. Index. Contents.