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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.

Internal elliptic regularity.


roposition

(Second order internal elliptic regularity). Let MATH be a bounded open set, MATH , MATH , MATH and MATH be a weak solution of the elliptic PDE (see the definition ( Elliptic differential operator )) MATH Then MATH and for any $V\subset U$ we have the estimate MATH where the constant $C$ depends only on $V,U$ and $L$ .

Proof

We follow the strategy described in the beginning of the section ( Elliptic regularity section ). Let MATH and $\zeta$ is the corresponding cutoff function for $V$ and $W$ , see the definition ( Cutoff function ). The weak solution $u$ is defined by the equation MATH We substitute MATH We have MATH We aim to use uniform positiveness of MATH . Hence, we transform the first term as follows: MATH Due to the presence of the cutoff function, the proposition ( Finite difference basics ) applies: MATH Hence, MATH We reverse the sign of all terms and estimate the left hand side from below using uniform positive definiteness of the matrix MATH : MATH We proceed to estimate the right hand side from above. Note that some of the terms have derivatives of second order. Such terms have to be estimated using the formula ( Cauchy inequality with epsilon ) with the epsilon to be much smaller than $\theta$ . Then presence such terms would not violate the estimate from below. For example, MATH Then the $\varepsilon~$ to be chosen to satisfy MATH . We perform such estimates for all terms on the right hand side and use the proposition ( Finite difference in Sobolev space ) to estimate the first order finite differences. We arrive to the estimate MATH Also, MATH We use the proposition ( Existence of derivative via finite difference ) to conclude MATH Finally, the estimate MATH follows from MATH with $v=\zeta^{2}u$ and similar calculations.

Proposition

(High order internal elliptic regularity). Let MATH be a bounded open set, MATH , MATH , MATH and MATH be a weak solution of the elliptic PDE (see the definition ( Elliptic differential operator )) MATH Then MATH and for any $V\subset U$ we have the estimate MATH where the constant $C$ depends only on $m,V,U$ and $L$ .

Proof

The proof is accomplished by induction in $m$ . For $m=0$ the statement is identical to the proposition ( Second order internal elliptic regularity ).

Assuming that the statement is true for $m$ we prove it for $m+1$ as follows.

Let $V\subset U$ . Let $\alpha$ is a multi-index (see the section ( Function spaces section )) and MATH . We set MATH in the identity MATH for some MATH . We integrate by parts and arrive to MATH where the $\tilde{f}$ is an expression containing derivatives of $u$ up to the order $m+2$ and derivatives of $f$ up to the order $m+1$ . Hence, by induction hypothesis, MATH and the we complete the induction using the proposition ( Second order internal elliptic regularity ).




Notation. Index. Contents.


















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