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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.
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.

Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.


et $P$ be a probability measure, MATH for a $\sigma $ -algebra $\QTR{cal}{F}$ . An $\QTR{cal}{F}$ -measurable random variable $X$ : MATH gives rise to the set function $\mu$ given by the relationship MATH The measure $\mu$ does not have to be positive or normalized to 1. This brings us to consideration of a wider class of measures.

Definition

The set $A$ is called "positive set of measure $\mu$ " if MATH for MATH .

Proposition

If MATH and $\mu$ is a $\sigma$ -additive measure then there exists a positive set $A_{+}\subseteq A$ .

Proof

We are not seeking to obtain a maximal in some sense positive set. The goal is obtain any positive set. Hence, any MATH -style procedure is not necessary.

If $A_{1}\equiv A$ does not have a subset of negative measure then we are done. Otherwise we can find a smallest integer such that MATH s.t. MATH (hence, we are trying to grab a set $B_{1}$ of negative measure with a large absolute value). We set MATH and repeat the procedure. If the procedure does not terminate at finding a positive set then we obtain a disjoint sequence MATH .

We check that MATH is a positive set. By construction, MATH is a non decreasing sequence. Since MATH then MATH . By construction of $B_{k}$ the set MATH does not have a set with negative measure greater then MATH by absolute value. This means that for any $\varepsilon$ there exists some MATH such that MATH does not have set with negative measure less then $-\varepsilon$ . Hence, the set MATH does not have a subset with negative measure.

Proposition

(Hahn decomposition theorem). Let $\mu$ be a signed measure. Then there exist a unique (up to measure-0 sets) pair of a positive set $\Omega_{+}$ and a negative set $\Omega_{-}$ such that MATH .

Proof

We seek a maximal positive set and its complement. Either there exists a set $A$ s.t. MATH or it exists not. In the later case MATH . In the former case there exists at least one positive set (see the previous proposition). Hence, the MATH is well defined. We select a sequence of positive sets MATH such that MATH . Then we take MATH . $\Omega_{+}$ is a positive set. The set MATH is a negative set. Indeed, existence of a set of positive measure within MATH leads to contradiction with the construction of $\Omega_{+}$ .

Definition

Measures $\mu$ and $\nu$ are called "mutually singular" iff MATH The measure $\nu$ is said to be "absolutely continuous" with respect to $\mu$ iff MATH

Proposition

(Jordan decomposition theorem). Let $\mu$ be a signed measure. The there exists a unique (up to sets of measure 0) pair of mutually singular positive measures $\mu_{+}$ and $\mu_{-}$ such that MATH .

Proof

Using the notation of the previous proposition set MATH and MATH .

Definition

The measure MATH is called "total variation" or "absolute value" of $\mu$ .

Proposition

(Radon-Nikodym theorem) Let $\nu,\mu$ are signed measures ( $\sigma$ -additive). Then $\nu$ is absolutely continuous with respect to $\mu$ if and only if MATH $\ $ for some positive measurable function $X$ . The function $X$ is unique.

Proof

The proposition contains several statements but the construction of $X$ based on $\nu$ and $\mu$ is the main idea and the only non-trivial part. For simplicity we assume that $\mu~$ is a positive measure (because otherwise we use the proposition ( Hahn decomposition theorem ) to modify the proof). We reconstruct the $X$ from values of $\nu$ and $\mu$ according to the following procedure. Observe that the measure $\nu-c\mu$ , $c\in\QTR{cal}{R}$ should be of the form MATH . We also have Hahn decomposition ( Hahn decomposition theorem ) at our disposal. Hence, we vary $c$ among the numbers MATH at consider the measures MATH . This gives us level sets of $X$ with increasing precision as $m\uparrow\infty$ . Then we construct an $m$ -indexed sequence of simple functions $X_{m}$ (see definition ( Simple function definition )). Let $\Omega_{\pm}^{c}$ are the Hahn decomposition sets of MATH . We set $X_{m}=c_{km}$ on MATH . Such sequence is almost surely a Cauchy sequence. Hence, it converges to some measurable integrable function $X$ . By the dominated convergence ( Dominated convergence theorem ) MATH . Then we estimate MATH for any $\varepsilon>0$ using the construction.

Notation

The function $X$ of the proposition ( Radon-Nikodym theorem ) is denoted MATH





Notation. Index. Contents.


















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