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.

## Elliptic regularity.

n this section we prove that if the function of the problem ( Elliptic Dirichlet problem ) is smooth and the boundary is then the solution is smooth. The general strategy is to start from the weak problem and substitute . Then we would integrate by parts in the and arrive to the situation of the form Where the "other terms" contain lower derivatives of . We would use the uniform positiveness of the matrix to conclude for some . Consequently, we estimate the right hand side from the above using standard means (such as the formula ( Cauchy inequality with epsilon )) and conclude .

The uniform positiveness of the matrix is a pointwise property. Hence, we cannot use weak derivatives and we are not given an apriori existence of . Therefore, we will be using finite difference approximations and passing it to the limit. Such operation requires some preliminaries.

 a. Finite differences in Sobolev spaces.
 b. Internal elliptic regularity.
 c. Boundary elliptic regularity.
 Notation. Index. Contents.