Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 $t,T$ be time moments, $t<T$ . The $\tau$ denotes the time of the first jump. We subdivide the interval $\left[ t,T\right] $ into $n$ small sub-intervals and let the size of each subinterval go to zero with $t,T$ being constants. We use the result MATH as $n\rightarrow\infty$ and for MATH and the notation MATH , MATH Observe that MATH We apply the ( Chain_rule ): MATH From point of view of MATH the events MATH are deterministic for $i=0,...,N-2$ , hence, MATH then, by ( Poisson property 1 ), MATH We continue in similar manner and take a limit: MATH




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

MATH (Poisson property 2)

Similarly, for any integer $k\geq0$ MATH where $\tau$ denotes any jump of $N_{t}$ and MATH denotes the set of permutations of $k$ integers from $1$ to $N$ . The factorial exists because permutations of MATH constitute identical terms in the sum while the equality is based on decomposition into disjoint events. We continue MATH MATH We note $N\Delta t=T-t$ , MATH = MATH , MATH , MATH . MATH Hence,

MATH (Poisson property 3)





Notation. Index. Contents.


















Copyright 2007