Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 MATH with smooth boundary and given functions MATH and MATH find a function MATH satisfying the relationships MATH where the operation $B$ is given by the definition ( Bilinear form B ).

We introduce the convenience notation MATH

Proposition

(Existence and uniqueness for penalized problem) If the coefficients of $B$ 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 $u_{1}$ and $u_{2}$ are solutions of the problem ( Stationary penalized problem ). Then MATH Note that MATH Then MATH and MATH follows from the condition ( Assumption of coercivity 1 ).

Problem

(Galerkin approximation of stationary problem) We construct finite dimensional spaces $S_{n}$ as follows.

Let $w_{0}\in K$ . We take the increasing set of linearly independent functions MATH with the properties MATH where we introduced MATH We pose the problem of finding a function $u_{n}\in S_{n}$ such that MATH

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 MATH and set consecutively MATH in the problem ( Galerkin approximation of stationary problem ). We end up with a system of equations MATH The condition ( Assumption of coercivity 1 ) implies that $\hat{B}$ is an invertible matrix and there is estimate MATH where the constant $\alpha$ is not dependent on $n$ . Let MATH then MATH We introduce the change of variable $y=x-x_{0}$ : MATH Note that the operation MATH has the following properties MATH for some constant $C$ independent of $n$ . Therefore, there is a value of the constant $c$ that makes the operation MATH a contraction. By the proposition ( Banach fixed point theorem ), for such $c$ there is a fixed point $y^{\ast}$ : MATH Suppose $c_{1}$ is such that the proposition ( Banach fixed point theorem ) is applicable so that MATH We aim to construct $y_{2}$ for a greater constant $c_{2}:$ MATH We subtract the above equalities and obtain MATH We make the change of variable $z=y_{1}-y_{2}$ : MATH The term MATH is a contraction with respect to $z$ if $c_{1}$ is chosen so that MATH with $C$ taken from MATH . The term MATH is a contraction if MATH is small enough. Hence, there is a solution $z$ according to the proposition ( Banach fixed point theorem ). We thus recover the value $y^{\ast}$ for all $c$ .

Proposition

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

1. MATH , for a constant $C$ depending only on $w_{0}$ , $f$ , coefficients of $L$ and $U$ .

2. MATH , for a constant $C$ depending only on on $w_{0}$ , $f$ , $\psi$ , coefficients of $L$ and $U$ .

Proof

(1) We set $v=u_{n}-w_{0}$ in the problem ( Galerkin approximation of stationary problem ): MATH and note that MATH : MATH The MATH has the property MATH , MATH : MATH We substitute MATH MATH We use the condition ( Assumption of coercivity 1 ), proposition ( Energy estimates for the bilinear form B ) and arrive to MATH hence MATH

Proof

(2) According to the relationship MATH , the last estimate and the proposition ( Energy estimates for the bilinear form B ) MATH We calculate MATH and note that MATH is nonnegative and $w_{0}\in K$ thus $w_{0}-\psi$ is nonpositive. Hence MATH and MATH according to MATH MATH Thus MATH or MATH

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 MATH , MATH such that MATH In addition, by the proposition ( Rellich-Kondrachov compactness theorem ) MATH and the operation MATH is continuous in MATH . Therefore MATH We pass the relationship MATH to the limit $n\rightarrow\infty$ and obtain MATH





Notation. Index. Contents.


















Copyright 2007