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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.
D. Maximum principles.
E. Eigenfunctions of symmetric elliptic operator.
F. Green formulas.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Energy estimates for bilinear form B.


roposition

(Energy estimates for the bilinear form B). Let $B$ be the bilinear form given by the definition ( Bilinear form B ) and MATH satisfy the definition ( Elliptic differential operator ). Then for any MATH MATH MATH where the constants $C_{1}>0$ , $C_{2}>0$ , $C_{3}\geq0$ are dependent only on the functions $a^{ij},b^{i},c$ and the set MATH .

Remark

We may add MATH to both sides and rewrite the second inequality as MATH

Proof

Based on the definition ( Bilinear form B ) of $B$ we directly estimate MATH Each of the integrals is dominated by MATH for some constant $C$ dependent only on $U$ . Hence, MATH

According to the definition ( Elliptic differential operator ) the matrix MATH is uniformly positive definite for all $x\in U$ . Hence, there exists a constant $\theta$ such that for any $x\in U$ MATH We integrate the above and use the definition ( Bilinear form B ) of $B$ : MATH

We use the formula ( Cauchy inequality with epsilon ) to estimate the second term: MATH and choose the $\varepsilon$ so that MATH We continue the estimation: MATH MATH where the constant MATH depends only on the functions $b^{i},c$ and the set $U$ .





Notation. Index. Contents.


















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