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.
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.

Fine structure of stopping time.


roposition

(Wiener-Hopf technique) Let MATH where the functions MATH are Fourier transforms of measures with support on MATH (or $[0,+\infty)$ ), the functions MATH are Fourier transforms of measures with support on $(-\infty,0]$ (or $(-\infty,0)$ respectively).

Suppose that there exists a $r_{0}>0$ such that the power series MATH converge for MATH and MATH then MATH

Proof

The equality MATH may be written as MATH We look at terms MATH for every $n$ : MATH

According to the proposition ( Inversion of ch.f. into p.m. 1 ), MATH where the $\mu_{p_{1}}$ is the measure that produces the Fourier transform $p_{1}$ . There is a similar statement for $q_{1},p_{1}^{\ast}$ and $q_{1}^{\ast}$ . We write the formula MATH in the form MATH and note that the LHS and RHS are produced from measures with disjoint supports. Hence, we must have MATH

With such result in place we proceed to the equation MATH and find MATH and so fourth: MATH

Proposition

(Ch.f. of entrance time 1) Let $\alpha$ be the time of first entrance into the interval $A$ with $A$ being one of the following: MATH , $[0,+\infty)$ , MATH , $(-\infty,0]$ : MATH

Then MATH

Proof

We prove the statement for the case MATH .

For an MATH , $t\in\QTR{cal}{R}$ and $f$ being the ch.f. function of $X$ we write MATH Using the proposition ( Ch.f. of a sum ) we substitute MATH : MATH and separate the sum into the pre- $\alpha$ and post- $\alpha$ parts MATH We calculate the first term: MATH We interchange the order of summation MATH : MATH where the MATH MATH is the Fourier transform of the measure MATH having support outside of MATH . Indeed, MATH implies that $S_{n}\not \in A$ .

We calculate the second term: MATH We make the change of summation index $n=\alpha+k$ : MATH By independence of pre- $\alpha$ and post- $\alpha$ fields, MATH The MATH is distributed like $S_{k}$ : MATH where the MATH : MATH is the Fourier transform of the measure MATH having support in MATH .

We put our results together: MATH MATH MATH We transform the $f\left( t\right) $ as follows. The Taylor decomposition of MATH at $x=0$ reads: MATH Therefore, MATH MATH The statement now follows from the proposition ( Wiener-Hopf technique ) and the formula MATH .

Until end of this section, the set $A$ and the r.v. $\alpha$ are defined as in the proposition ( Ch.f. of entrance time 1 ).

Proposition

We have MATH and MATH iff MATH in which case MATH

Proposition

Suppose that $X\not \equiv 0$ and at least one of MATH is finite, then MATH

Proposition

If MATH and MATH then MATH

Proposition

Suppose that $X\not \equiv 0$ and at least one of MATH is finite. Let MATH , MATH .

1. If MATH but may be $+\infty$ then MATH .

2. If MATH then MATH are both finite iff MATH .





Notation. Index. Contents.


















Copyright 2007