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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
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.
A. Energy estimates for bilinear form B.
B. Existence of weak solutions for elliptic Dirichlet problem.
C. Elliptic regularity.
a. Finite differences in Sobolev spaces.
b. Internal elliptic regularity.
c. Boundary elliptic regularity.
D. Maximum principles.
E. Eigenfunctions of symmetric elliptic operator.
F. Green formulas.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Finite differences in Sobolev spaces.


efinition

(Finite differences). For a locally summable function MATH , we introduce the notations MATH Here the $e_{i}$ is the $i$ -th coordinate vector, MATH .

Definition

(Cutoff function). Let MATH . The cutoff function is any function MATH that satisfies the following conditions: MATH

The importance of the cutoff function is evident from the following propositions. Note, that we have to restrict the original set $U$ to a subset $V$ .

Proposition

(Finite difference in Sobolev space). Let $1\leq p<\infty$ and MATH . Then for any subset $V\subset U$ MATH for $h$ such that MATH dist MATH .

Proof

Due to the proposition ( Local approximation by smooth functions ) it is enough to prove the statement for a smooth $u$ . We have MATH Hence, for $1\leq p<\infty$ MATH MATH

Proposition

(Finite difference basics). Let MATH be a bounded set and $v,u$ are locally summable functions. Then MATH If, in addition, MATH then

MATH (Integration by part for finite differences)

Proof

We verify the statements as follows: MATH MATH

Proposition

(Existence of derivative via finite difference). Let $1<p<\infty$ , MATH , $V\subset U$ and MATH for some constant $C$ and any $h$ such that MATH dist MATH . Then MATH

Proof

According to the proposition ( Weak compactness of bounded set ) there exists a sequence MATH , $h_{k}\downarrow0$ such that MATH for every $i=1,...,n$ . Then we pass the formula ( Integration by part for finite differences ) to the limit and use the definition ( Weak derivative ) to establish that MATH in the weak sense. Then MATH follows from MATH .





Notation. Index. Contents.


















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