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.

## Elliptic PDE.

et be an open bounded subset of . The are functions . The is a linear differential operator

 (Operator L)

Problem

(Elliptic Dirichlet problem).

Definition

(Elliptic differential operator). The operator is called "elliptic" if the matrix is uniformly positive definite for all .

Definition

(Bilinear form B). Let , and . We introduce the notation

Let us multiply the equations of the problem ( Elliptic Dirichlet problem ) with a function and integrate with respect to over . After integration by parts we obtain

Definition

(Weak solution of elliptic Dirichlet problem). The function is a "weak solution" of the problem ( Elliptic Dirichlet problem ) if it satisfies

 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.
 Notation. Index. Contents.