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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.

Random walk.


e consider the sequence of r.v. MATH and a triple MATH . The $\Omega$ is the event space of MATH and $\QTR{cal}{F}$ is the minimal $\sigma $ -algebra that makes all MATH measurable. We think of $X_{n}$ as a consecutive sequence of trials as $n$ increases. Hence, MATH is a "process" and $n$ is a time parameter.

We use the notation $\QTR{cal}{F}_{n}$ to represent the $\sigma$ -algebra containing the information available up to time $n$ . Hence, $\QTR{cal}{F}_{n}$ is the minimal $\sigma$ -algebra that makes the family MATH measurable.

We use the notation MATH to represent the $\sigma $ -algebra containing the information that comes after the time $n$ . Hence, MATH is the minimal $\sigma$ -algebra that makes the the family MATH measurable.

We use the notation MATH to represent the minimal $\sigma $ -algebra that contains all of the $\QTR{cal}{F}_{n}$ , $n=1,2,...$ . By the minimality of $\QTR{cal}{F}$ we have MATH

Proposition

(Random walk space approximation). Given $\varepsilon>0$ and MATH there exists MATH such that MATH

We use the notation MATH for the intersection

MATH (Remote field)

Sometimes it is convenient to think of $\Omega\,$ as the product space

MATH (Random walk space)
where each $\Omega_{n}$ is the probability space for $X_{n}$ . The $P$ is the product measure consistent with the d.f. $F_{n}$ on each $\Omega_{n}$ . The $\Omega$ is a collection of infinite sequences of real numbers: MATH The $\tau$ denotes the "shift":
MATH (Shift)
The $\tau^{k}$ is the $\tau$ applied $k$ times.

The MATH denotes the set of permutations of $n$ integers from the range $k,...,n$ . A permutation MATH (and similarly MATH ) produces a mapping on MATH and on $\Omega$ according to the rules MATH

MATH (Permutation)

Definition

(Invariant set) The set MATH is called "invariant" if MATH .

Definition

(Permutable set) The set MATH is called "permutable" if $\pi\Lambda=\Lambda$ for every permutation of finite number of positions. A function MATH is "permutable" if MATH .

Definition

(Remote event). Any set MATH (see the formula ( Remote field )) is called "remote event".

Clearly, an invariant set is remote and a remote set is permutable.

Definition

(Independent process) The family MATH is an "independent" process if MATH are independent r.v. The family MATH is "stationary independent" process if MATH are iid.

Definition

(Random walk) For a stationary independent process MATH we defined the "random walk" as the process MATH , where $S_{0}=0$ and MATH for $j=1,2,...$ .




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.

Notation. Index. Contents.


















Copyright 2007