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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.
 a. Cauchy and Young inequalities.
 b. Cauchy inequality for scalar product.
 c. Holder inequality.
 d. Lp interpolation.
 e. Chebyshev inequality.
 f. Lyapunov inequality.
 g. Jensen inequality.
 h. Estimate of mean by probability series.
 i. Gronwall inequality.
 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.

## Holder inequality.

roposition

Let are real numbers connected by the relationship and , . Then

 (Holder inequality)

Proof

The formula ( Young inequality ) implies Set , where and then Hence,

Proposition

Assume that is a finite collection of real numbers connected by the relationship and . Then

 (Holder inequality 2)

Proof

The proposition is a consequence of the formula ( Holder inequality ) by induction.

Proposition

Let be a positive function. Assuming that all the integrals are well defined and are connected as above, the following estimate holds

 (Holder inequality 3)

Proof

We essentially repeat the proof of the formula ( Holder inequality ). Set , where and then Hence,

 Notation. Index. Contents.