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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.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
E. Various types of convergence.
a. Uniform convergence and convergence almost surely. Egorov's theorem.
b. Convergence in probability.
c. Infinitely often events. Borel-Cantelli lemma.
d. Integration and convergence.
e. Convergence in Lp.
f. Vague convergence. Convergence in distribution.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. 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.

Infinitely often events. Borel-Cantelli lemma.


efinition

(Limsup and liminf for sets) MATH ; MATH ;

Proposition

MATH if and only if $\omega$ belongs to infinitely many sets $A_{n}$ . MATH if and only if it belong to all $A_{n}$ starting from some $n$ .

Proof

We use the technique of the section ( Operations on sets and logical statements ). The statement MATH means exactly MATH The statement MATH means exactly MATH

Definition

MATH . The "i.o." stands for "infinitely often".

Proposition

(Borel-Cantelli lemma, part 1). MATH MATH .

Proof

The sequence of sets MATH is decreasing as $m\uparrow$ . Hence, MATH . Consequently, MATH because MATH and MATH

Proposition

(Borel-Cantelli lemma, part 2). If the sets MATH are independent ( MATH , $i\not =j$ ) then MATH .

Proof

Note that MATH . Hence, consider MATH By independence, MATH Every term in the above sum is zero because MATH .

Proposition

(IO criteria for AS convergence) For a sequence of r.v. MATH the following are equivalent statements:

1. $X_{n}\rightarrow0$ a.s.

2. MATH , MATH

Proof

According to the section ( Operations on sets and logical statements ) the $X_{n}\rightarrow0$ a.s. if the following set $A$ has the complement of probability zero: MATH Observe that the composition MATH is the " MATH i.o." and since we take the intersection in $m$ , the set " MATH i.o." has the complement of probability zero for every $m$ . This argument works in either direction.





Notation. Index. Contents.


















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