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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.

Existence of weak solution for parabolic Dirichlet problem.


roposition

(Existence of weak solution for the parabolic Dirichlet problem). There exists a weak solution of the problem ( Parabolic Dirichlet problem ).

Proof

The sequence MATH of approximate Galerkin solutions given by the equations ( Galerkin problem ) satisfies the relationships MATH for any $v$ from the linear span of MATH , $N\leq m$ . According to the estimates of the proposition ( Energy estimates for the Galerkin approximate solution ) and the proposition ( Weak compactness of bounded set ) there is a subsequence of MATH that converges weakly in MATH to some MATH and MATH converges weakly in MATH . Such convergence is what we need to pass to the limit in (*) and obtain MATH for any MATH .

Proposition

(Uniqueness of weak solution for the parabolic Dirichlet problem). The weak solution of the problem ( Parabolic Dirichlet problem ) is unique.

Proof

Set $v=u$ in the definition ( Weak solution of parabolic Dirichlet problem ) with $f=g=0$ and use the proposition ( Energy estimates for the bilinear form B ) and the proposition ( Differential inequality 2 ).





Notation. Index. Contents.


















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