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 be a bounded open set, , , and be a weak solution of the elliptic PDE (see the definition ( Elliptic differential operator )) Then and for any we have the estimate where the constant depends only on and .

Proof

We follow the strategy described in the beginning of the section ( Elliptic regularity section ). Let and is the corresponding cutoff function for and , see the definition ( Cutoff function ). The weak solution is defined by the equation We substitute We have We aim to use uniform positiveness of . Hence, we transform the first term as follows: Due to the presence of the cutoff function, the proposition ( Finite difference basics ) applies: Hence, We reverse the sign of all terms and estimate the left hand side from below using uniform positive definiteness of the matrix : 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 . Then presence such terms would not violate the estimate from below. For example, Then the to be chosen to satisfy . 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 Also, We use the proposition ( Existence of derivative via finite difference ) to conclude Finally, the estimate follows from with and similar calculations.

Proposition

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

Proof

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

Assuming that the statement is true for we prove it for as follows.

Let . Let is a multi-index (see the section ( Function spaces section )) and . We set in the identity for some . We integrate by parts and arrive to where the is an expression containing derivatives of up to the order and derivatives of up to the order . Hence, by induction hypothesis, and the we complete the induction using the proposition ( Second order internal elliptic regularity ).

 Notation. Index. Contents.