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.

## Finite differences in Sobolev spaces.

efinition

(Finite differences). For a locally summable function , we introduce the notations Here the is the -th coordinate vector, .

Definition

(Cutoff function). Let . The cutoff function is any function that satisfies the following conditions:

The importance of the cutoff function is evident from the following propositions. Note, that we have to restrict the original set to a subset .

Proposition

(Finite difference in Sobolev space). Let and . Then for any subset for such that dist .

Proof

Due to the proposition ( Local approximation by smooth functions ) it is enough to prove the statement for a smooth . We have Hence, for

Proposition

(Finite difference basics). Let be a bounded set and are locally summable functions. Then If, in addition, then

 (Integration by part for finite differences)

Proof

We verify the statements as follows:

Proposition

(Existence of derivative via finite difference). Let , , and for some constant and any such that dist . Then

Proof

According to the proposition ( Weak compactness of bounded set ) there exists a sequence , such that for every . Then we pass the formula ( Integration by part for finite differences ) to the limit and use the definition ( Weak derivative ) to establish that in the weak sense. Then follows from .

 Notation. Index. Contents.