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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.

Integration and convergence.


efinition

"Simple random variable" (simple function) is a r.v. of the form MATH where the $1_{A}$ is the indicator function of the event (set) $A$ : MATH and MATH are scalars.

Definition

The expectation (integral) of a simple random variable is MATH For any positive random variable $X$ we define the integral as MATH where the $\sup$ is taken over all simple positive r.v. $\xi$ such that MATH for MATH .

If the $\sup$ is finite then the r.v. (function of $\omega$ ) is called "summable" on $A$ .

Proposition

(Fatou lemma) If MATH is a sequence of non-negative random variables and $X_{n}\rightarrow X$ a.s. on $A$ then MATH

Proof

It is sufficient to show that for any simple r.v. $\xi$ such that $\xi\leq X$ on $A$ we have MATH for any small $\varepsilon$ and sufficiently large $n$ .

We start by writing definition of a.s. convergence according to the recipes of the section ( Operations on sets and logical statements ) MATH up to a set of measure 0. We fix some $m_{0}$ then MATH Let's introduce the notation MATH By positiveness of the $X_{n}$ and $X$ we have MATH for any simple function $\xi$ , $\xi<X$ . The $A_{N}$ is an increasing sequence, hence, by the proposition ( Continuity lemma ) MATH Pick MATH such that MATH . For MATH we have MATH The last two terms are arbitrarily small and the statement holds for sufficiently large $n$ and any simple r.v. $\xi<X$ .

Proposition

(Dominated convergence theorem) Let MATH is such that $X_{n}\rightarrow X$ a.s. on $A$ , MATH and $\int YdP<\infty$ . Then MATH .

Proof

Apply Fatou lemma to MATH and MATH .





Notation. Index. Contents.


















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