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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.

Energy estimates for Galerkin approximate solution.


roposition

(Energy estimates for the Galerkin approximate solution). The solution $u_{m}$ of the equations ( Galerkin problem ) satisfies the estimates MATH for all $m$ and constants $C_{1},C_{2},C_{3}$ depending only on $U,T$ and the functions MATH .

Proof

We multiply the equations ( Galerkin problem ) by MATH and sum for $k=1,...,m$ . We obtain MATH We add the relationship MATH to the second estimate of the proposition ( Energy estimates for the bilinear form B ): MATH and obtain MATH We combine the above with (*) and obtain MATH We apply the formula ( Cauchy inequality ) to the term MATH : MATH

We drop the term MATH in the formula (**), recall the starting condition for MATH at $t=0$ : MATH and apply the proposition ( Differential inequality 2 ): MATH Thus, we proved the first desired inequality.

Next, we drop the term MATH from the formula (**) and integrate in $t$ : MATH hence, we proved the second inequality.

To prove the final inequality, we seek to estimate the quantity MATH

for any MATH such that MATH See the remark in the section ( Dual Sobolev spaces ).

We introduce $P^{m}$ : the projection on the linear span of MATH in MATH . We deduce from the formula ( Galerkin problem ) that MATH and apply the formula ( Holder inequality ) and the proposition ( Energy estimates for the bilinear form B ) to estimate the RHS: MATH Note that MATH , hence MATH We integrate the last inequality in $t$ and use the first inequality of this proposition MATH





Notation. Index. Contents.


















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