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 for backward Kolmogorov equation.

e adapt derivations of the section ( Discretization with respect to time parameter ) to backward Kolmogorov equation.

Condition

(Backward Kolmogorov PDE setup) We adopt the setup ( Generic parabolic PDE setup ) with the following exception. We are considering the problem

 (Backward Kolmogorov PDE)
for some and

The evolution of the solution happens in opposite direction: from at time to time . We adopt notation and constructs of the sections ( Weak formulation with respect to time parameter ) and ( Discretization with respect to time parameter ). Thus, we introduce the following definitions.

Definition

(Class ) We introduce the class of functions

For any , we apply the operation to the equation ( Backward Kolmogorov PDE ) and integrate over : We integrate the first term by parts:

Problem

(Weak backward Kolmogorov with respect to time) In context of the condition ( Backward Kolmogorov PDE setup ) we seek such that

 (Time-weak backward Kolmogorov)
for any .

We redefine the partition to reflect backward evolution with time:

Definition of the class remains the same: We introduce the notations for .

We calculate for and : Note that We continue

The equation ( Time-weak backward Kolmogorov ) transforms into for any . Thus, . We replace with : We merge conditions and arrive to We fix and take a with support within . We arrive to the following problem.

If then If then

Problem

(Backward discontinuous Galerkin time-discretization) In context of the condition ( Backward Kolmogorov PDE setup ) we seek such that for every and any ,

Note that the solution of the equation ( Backward Kolmogorov PDE ) satisfies the above problem.

 Notation. Index. Contents.