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.
 6 Variational inequalities.
 A. Stationary variational inequalities.
 a. Weak and strong formulations for stationary variational inequality problem.
 b. Existence and uniqueness for coercive stationary problem.
 c. Penalized stationary problem.
 d. Proof of existence for stationary problem.
 e. Estimate of penalization error for stationary problem.
 f. Monotonicity of solution of stationary problem.
 g. Existence and uniqueness for non-coercive stationary problem.
 B. Evolutionary variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Estimate of penalization error for stationary problem. roposition

(Penalization error) Assume that

1. the coefficients of satisfy the definition ( Elliptic differential operator ),

2. the condition ( Assumption of coercivity 1 ) holds,

3. ,

4. ,

5. on then for the solution of the problem ( Stationary penalized problem ) and the solution of the problem ( Stationary variational inequality problem ) the following estimate holds Proof

We set in the problem ( Stationary penalized problem ): and note that thus we obtain or We estimate the RHS using the propositions ( Energy estimates for the bilinear form B ) and ( Cauchy inequality ) and we estimate the LHS using the condition ( Assumption of coercivity 1 ). We obtain Using such estimate we revisit , use the same tools again and obtain We introduce the quantity Thus it remains to show that Note that . Indeed, also, since and on  We set in the problem ( Stationary penalized problem ) and set in the problem ( Stationary variational inequality problem ): We add and and obtain Hence But hence  Thus and by the condition ( Assumption of coercivity 1 ) and the propositions ( Energy estimates for the bilinear form B )-1, Together with this implies the statement of the proposition.

 Notation. Index. Contents.