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.

## Galerkin approximation for parabolic Dirichlet problem. e are considering the problem ( Parabolic Dirichlet problem ).

Let is the basis of formed by the eigenfunctions of the operator with the following normalization: (Definition of Galerkin basis 1)
The existence of such basis follows from the proposition ( Eigenvalues of symmetric elliptic operator ). To derive the orthogonality in , write and integrate by parts on the LHS.

We form the sum and select the functions to satisfy the conditions We integrate by parts the term and write the equations for in the form (Galerkin problem)
We substitute the definition of and use the orthonormality of . We obtain (Galerkin coefficients)
This is a Cauchy system of ODEs and it has a form that insures that there is always a solution .

 Notation. Index. Contents.