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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.
15. Parabolic PDE.
A. Galerkin approximation for parabolic Dirichlet problem.
B. Energy estimates for Galerkin approximate solution.
C. Existence of weak solution for parabolic Dirichlet problem.
D. Parabolic regularity.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Parabolic PDE.


et $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ . The MATH are functions MATH . The $T$ is a positive number. The $L$ is a linear differential operator

MATH (Operator L 2)

Problem

(Parabolic Dirichlet problem). MATH

Definition

(Parabolic differential operator). The operator MATH is called "parabolic" if the matrix MATH is uniformly positive definite for all MATH : MATH

Definition

(Bilinear form B 2). Let MATH , MATH and MATH . We introduce the notation MATH

Let us multiply the equations of the problem ( Parabolic Dirichlet problem ) with a function MATH and integrate with respect to $x$ over $U$ . After integration by parts we obtain MATH

Definition

(Weak solution of parabolic Dirichlet problem). The function MATH is a "weak solution" of the problem ( Parabolic Dirichlet problem ) if it satisfies MATH

To understand the requirement MATH observe that MATH with MATH and compare with the propositions ( Representation of dual Sobolev space ) and ( Embedding of dual Sobolev space ). We see that the expression MATH is bounded for MATH : MATH MATH Finally MATH and MATH hence we need MATH




A. Galerkin approximation for parabolic Dirichlet problem.
B. Energy estimates for Galerkin approximate solution.
C. Existence of weak solution for parabolic Dirichlet problem.
D. Parabolic regularity.

Notation. Index. Contents.


















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