Quantitative Analysis
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Numerical Analysis
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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.

Optional random variable. Stopping time.


efinition

(Optional random variable) A r.v. $\alpha$ is called "optional" relative to the stochastic process MATH if it takes values in MATH and satisfies the condition MATH where the $\QTR{cal}{F}_{n}~$ is the $\sigma$ -algebra generated by the MATH .

The $\alpha$ points out some time index and does so based on the information about the path of the process up to that time index. For a particular path $\omega$ one can have MATH for at most one index: MATH For these reasons the $\alpha$ is sometimes called "stopping time".

Definition

(Pre- $\alpha$ field) The pre- $\alpha$ field MATH , MATH , is the collection MATH

The pre- $\alpha$ field is a collection of scenarios that lead to the optional event MATH . In particular, MATH

Definition

(Post- $\alpha$ process) The post- $\alpha$ process MATH is defined on the event MATH using the relationship MATH

The post- $\alpha$ field MATH is the field generated by the post- $\alpha$ process.

Thus, for every MATH the time MATH is defined and then MATH for $n=1,2,...$ . In other words, $X_{\alpha+n}$ is the part of the path MATH after the stopping time $\alpha$ with the pre- $\alpha$ part shifted forward and away: MATH

Proposition

(Independency of pre-alpha and post-alpha fields) For a stationary independent process and an almost everywhere finite optional r.v. $\alpha$ related to it, the pre- $\alpha$ and post- $\alpha$ fields are independent. Furthermore, the post- $\alpha$ process is a stationary independent process with the same common distribution as the original one.

Proof

We seek to prove that MATH The statement assumes that $\alpha$ is almost everywhere finite: MATH Hence, we calculate MATH It follows from the definition of MATH that MATH . Therefore, we continue MATH We note that the events MATH and MATH belong to the fields $\QTR{cal}{F}_{n}$ and MATH on the opposite sides of the time index $n$ . We use the independence of MATH : MATH We can use the same trick to show that the events MATH are independent. Hence, MATH

Definition

(AlphaK and BetaK 1) We introduce the following random variables $\alpha^{k}$ and $\beta_{k}$ : MATH

Assuming the representation of the formula ( Random walk space ), MATH MATH In other words, $\alpha^{2}$ is the stopping time calculated from the path $\omega^{1}$ obtained by shifting forward the original path $\omega$ by $\alpha$ calculated on $\omega$ : MATH and we continue MATH The MATH is the path obtained by shifting forward $\omega_{1}$ by the $\alpha$ calculated on $\omega^{1}.$ We call it $\omega^{2}$ : MATH Hence, we obtain another way to understand $\alpha^{k}$ and $\beta_{k}$ .

Definition

(AlphaK and BetaK 2) We introduce the random variables $\alpha^{k}$ according to the rules MATH The variable $\beta_{k}$ is defined by the property MATH in other words, the data at the first position in $\omega^{k}$ is the data at $\beta_{k}+1$ -th position in the $\omega$ .

Proposition

(BetaK separation of random walk). Let MATH be a stationary independent process. Then

1. The random vectors MATH MATH are iid.

2. For a Borel measurable function $\phi$ , the r.v. MATH MATH are iid.

Proof

Assuming the representation of the formula ( Random walk space ), MATH MATH

Note that MATH points at the slot MATH of MATH . Hence, $V_{k}$ depends only on the part of the path $\omega$ between the $\beta_{k-1}+1$ -th position and the $\beta_{k}$ -the position and the manner of the dependency is the same for all $k$ .

Proposition

(Maximum of random walk) Let MATH be a stationary independent process, MATH is the associated random walk and the r.v. $\alpha,M$ are defined by MATH Then the statements A,B,C are equivalent and the statements a,b,c are equivalent.

A. MATH .

B. MATH .

C. MATH .

a. MATH .

b. MATH .

c. MATH .

Proof

Note that the events MATH and MATH are permutable events. Therefore, according to the proposition ( Hewitt and Savage zero-or-one law ) the values 1,0 are the only possible values for MATH and MATH . Hence, if we prove equivalence of A,B,C then we also obtain equivalence of a,b,c.

We prove MATH as follows. According to the proposition ( BetaK separation of random walk ), the variables MATH are iid. Hence, the proposition ( Strong law of large numbers for iid r.v. ) applies and we derive MATH where MATH . We also have MATH Hence, MATH Also, MATH thus MATH This implies B.

By definition of $M$ , MATH hence, B implies C. The C implies A by definitions of $\alpha$ and $M$ .

Proposition

(Eventuality of random walk) Let MATH be a stationary independent process and MATH is the associated random walk. There are only four mutually exclusive possibilities, each taking place a.s.

1. $\forall n:S_{n}=0$ ,

2. MATH ,

3. MATH ,

4. MATH .

Proof

If $X_{n}=0$ then (1) takes place. We exclude such possibility from further consideration. The MATH is a permutable r.v., hence, by the proposition ( Hewitt and Savage zero-or-one law ) it is a constant $c$ almost surely. Note that MATH thus MATH Therefore, either MATH or MATH .





Notation. Index. Contents.


















Copyright 2007