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.

## Penalized stationary problem.

roblem

(Stationary penalized problem) For a bounded set with smooth boundary and given functions and find a function satisfying the relationships where the operation is given by the definition ( Bilinear form B ).

We introduce the convenience notation

Proposition

(Existence and uniqueness for penalized problem) If the coefficients of satisfy the definition ( Elliptic differential operator ) and the condition ( Assumption of coercivity 1 ) then the problem ( Stationary penalized problem ) has a unique solution.

Proof

(Uniqueness) Assume that and are solutions of the problem ( Stationary penalized problem ). Then Note that Then and follows from the condition ( Assumption of coercivity 1 ).

Problem

(Galerkin approximation of stationary problem) We construct finite dimensional spaces as follows.

Let . We take the increasing set of linearly independent functions with the properties where we introduced We pose the problem of finding a function such that

Proposition

(Existence for Galerkin approximation of stationary problem) Under condition ( Assumption of coercivity 1 ) there exists a solution of the problem ( Galerkin approximation of stationary problem ).

Proof

We look for and set consecutively in the problem ( Galerkin approximation of stationary problem ). We end up with a system of equations The condition ( Assumption of coercivity 1 ) implies that is an invertible matrix and there is estimate where the constant is not dependent on . Let then We introduce the change of variable : Note that the operation has the following properties for some constant independent of . Therefore, there is a value of the constant that makes the operation a contraction. By the proposition ( Banach fixed point theorem ), for such there is a fixed point : Suppose is such that the proposition ( Banach fixed point theorem ) is applicable so that We aim to construct for a greater constant We subtract the above equalities and obtain We make the change of variable : The term is a contraction with respect to if is chosen so that with taken from . The term is a contraction if is small enough. Hence, there is a solution according to the proposition ( Banach fixed point theorem ). We thus recover the value for all .

Proposition

(A priory estimates for Galerkin solution 1) Let be a solution of the problem ( Galerkin approximation of stationary problem ) in dimensions. If the coefficients of satisfy the definition ( Elliptic differential operator ) and the condition ( Assumption of coercivity 1 ) then the following estimates hold

1. , for a constant depending only on , , coefficients of and .

2. , for a constant depending only on on , , , coefficients of and .

Proof

(1) We set in the problem ( Galerkin approximation of stationary problem ): and note that : The has the property , : We substitute We use the condition ( Assumption of coercivity 1 ), proposition ( Energy estimates for the bilinear form B ) and arrive to hence

Proof

(2) According to the relationship , the last estimate and the proposition ( Energy estimates for the bilinear form B ) We calculate and note that is nonnegative and thus is nonpositive. Hence and according to Thus or

Proof

We now prove the existence part of the proposition ( Existence and uniqueness for penalized problem ).

According to the propositions ( A priory estimates for Galerkin solution 1 ) and ( Weak compactness of bounded set ) there are functions , such that In addition, by the proposition ( Rellich-Kondrachov compactness theorem ) and the operation is continuous in . Therefore We pass the relationship to the limit and obtain

 Notation. Index. Contents.