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.

## Proof of existence for stationary problem.

e prove existence part of the proposition ( Existence and uniqueness for stationary problem ).

According to the proposition ( A priory estimates for Galerkin solution 1 )-1 and ( Weak compactness of bounded set ) there is a sequence such that Then by the proposition ( Rellich-Kondrachov compactness theorem ), According to the proposition ( A priory estimates for Galerkin solution 1 )-2 so that We cannot immediately pass to the limit in because of the term in . Hence we write and, assuming that , , Since , we now get rid of the problem term: and pass to the limit

 Notation. Index. Contents.