Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.
 15 Parabolic PDE.
 A. Galerkin approximation for parabolic Dirichlet problem.
 B. Energy estimates for Galerkin approximate solution.
 C. Existence of weak solution for parabolic Dirichlet problem.
 D. Parabolic regularity.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

Parabolic PDE.

et be a bounded subset of . The are functions . The is a positive number. The is a linear differential operator

 (Operator L 2)

Problem

(Parabolic Dirichlet problem).

Definition

(Parabolic differential operator). The operator is called "parabolic" if the matrix is uniformly positive definite for all :

Definition

(Bilinear form B 2). Let , and . We introduce the notation

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

Definition

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

To understand the requirement observe that with and compare with the propositions ( Representation of dual Sobolev space ) and ( Embedding of dual Sobolev space ). We see that the expression is bounded for : Finally and hence we need

 A. Galerkin approximation for parabolic Dirichlet problem.
 B. Energy estimates for Galerkin approximate solution.
 C. Existence of weak solution for parabolic Dirichlet problem.
 D. Parabolic regularity.
 Notation. Index. Contents.