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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
A. Zero-or-one laws.
B. Optional random variable. Stopping time.
C. Recurrence of random walk.
D. Fine structure of stopping time.
E. Maximal value of random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
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.

Maximal value of random walk.


e will be using the notation of the proposition ( Maximum of random walk ).

Notation

For a random walk MATH we introduce the following quantities: MATH The $\rho_{n}$ is understood as a permutation acting on $\omega$ .

Proposition

(Spitzer identity)

1. We have for MATH : MATH

2. $M$ is finite a.s. iff MATH in which case we have MATH

Proof

(1) We calculate MATH Note that MATH . Indeed, the MATH says that among positions MATH the maximum is attained at $k$ -th and the MATH says that among positions MATH the maximum is attained at $k$ . Hence, we continue MATH Note that MATH and MATH belong to pre- $k$ and post- $k$ event fields, hence, these are independent: MATH The MATH is distributed as $L_{n-k}$ : MATH We write the expression of interest and substitute the latest result: MATH The last expression has the structure MATH . One may see by concentrating at each $r^{n}$ term that such structure is the product MATH . Hence, MATH We calculate each term of the formula MATH . MATH Note that MATH . Also, note that MATH . Hence, MATH . We continue: MATH At this point we invoke the proposition ( Ch.f. of entrance time 1 ): MATH with $\alpha$ substituted for $\beta$ and $A=(-\infty,0]$ and obtain MATH We calculate the second term of the formula MATH similarly, using MATH : MATH We collect the results: MATH

Proposition

MATH

Notation

We introduce the r.v. MATH and MATH as follows MATH

Proposition

For each MATH the random vectors MATH and MATH have the same distribution.

For each MATH the random vectors MATH and MATH have the same distribution.

Proposition

For MATH MATH

Proposition

If the common distribution of stationary independent process is symmetric then MATH





Notation. Index. Contents.


















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