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.

## Distribution of Poisson process.

he purpose of this section is to compute some basic probabilities associated with Poisson process. Let be time moments, . The denotes the time of the first jump. We subdivide the interval into small sub-intervals and let the size of each subinterval go to zero with being constants. We use the result as and for and the notation , Observe that We apply the ( Chain_rule ): From point of view of the events are deterministic for , hence, then, by ( Poisson property 1 ), We continue in similar manner and take a limit:

Therefore, with use of the formula ( Poisson property 1a ), we conclude

 (Poisson property 2)

Similarly, for any integer where denotes any jump of and denotes the set of permutations of integers from to . The factorial exists because permutations of constitute identical terms in the sum while the equality is based on decomposition into disjoint events. We continue We note , = , , . Hence,

 (Poisson property 3)

 Notation. Index. Contents.