I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 A. Change of variables for parabolic equation.
 B. Discontinuous Galerkin technique.
 a. Weak formulation with respect to time parameter.
 b. Discretization with respect to time parameter.
 c. Discretization for backward Kolmogorov equation.
 d. Existence and uniqueness for time-discretized problem.
 e. Convergence of discontinuous Galerkin technique. Adaptive time stepping.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Discretization with respect to time parameter.

n context of the condition ( Generic parabolic PDE setup ) we introduce partition of the interval : We define to be the class We introduce the notations for .

We aim to approximate the solution of the problem ( Weak formulation with respect to time parameter ) with a function from for fixed and vanishing . We calculate for and : Note that We continue The equation ( Time-weak problem ) transforms into for any . Thus, . We replace with : We can switch former problem for the latter if we can prove later that both problems have a unique solution. We merge conditions and arrive to At this point we fix and take a with support within . We arrive to the following problem.

Problem

(Discontinuous Galerkin time-discretization) In context of the condition ( Generic parabolic PDE setup ) we seek such that for every and any ,

Note that the solution of the equation ( Generic parabolic PDE problem ) satisfies the above problem.

 Notation. Index. Contents.