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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.
B. Evolutionary variational inequalities.
a. Strong and variational formulations for evolutionary problem.
b. Existence and uniqueness for evolutionary problem.
c. Penalized evolutionary problem.
d. Proof of existence for evolutionary problem.
VIII. Bibliography
Notation. Index. Contents.

Penalized evolutionary problem.


roblem

(Evolutionary penalized problem) For a bounded set MATH with smooth boundary, time interval $\left[ 0,T\right] $ and given functions MATH , MATH find a function MATH satisfying the relationships MATH where the operation $B$ is given by the definition ( Bilinear form B 2 ).

Remark

(Preliminary reduction 2) In the section ( Convergence of finite elements for generic parabolic operator section ) we saw how a change of the function MATH transforms the problem to insure that the condition ( Assumption of coercivity 1 ) is met. In particular, if MATH then MATH and in light of the proposition ( Energy estimates for the bilinear form B )-2 the $\kappa$ may be chosen so that the bilinear form MATH would satisfy the condition ( Assumption of coercivity 1 ).

Proposition

(Existence and uniqueness for penalized problem 2) If the coefficients of $B$ satisfy the definition ( Elliptic differential operator ), the condition ( Assumption of coercivity 1 ) is satisfied and $K\not =\not 0 $ then the problem ( Evolutionary penalized problem ) has a unique solution.

Proof

(Uniqueness) Suppose $u_{1}$ and $u_{2}$ are two solutions of the problem ( Evolutionary penalized problem ). Then we deduce from the problem MATH We introduce the notation $w=u_{1}-u_{2}$ , subtract and put $v=w$ MATH Since MATH we derive MATH and conclude the uniqueness as in the proof of the proposition ( Existence and uniqueness for evolutionary problem ).

Proof

(Existence) We define $w_{0}$ to be the solution of MATH and define the sequence MATH as follows MATH We denote MATH and derive from MATH : MATH We add and obtain MATH or MATH We integrate both sides over $\left[ t,T\right] $ : MATH or MATH Note that MATH hence MATH Therefore MATH Let MATH is the point where the MATH (or we would modify the argument for a sequence that approaches the $\max$ ). Then considering $\left( \&\right) $ at $\tau$ we obtain MATH Thus MATH or MATH Thus, for $T-t$ sufficiently small MATH But then, using such result MATH and MATH we increase the $T-t$ and expand it to $\left[ 0,T\right] $ . Thus MATH for some $w$ . Then from MATH we also conclude MATH Then MATH implies that MATH and by passing MATH to the limit, $w$ is the solution of the problem ( Evolutionary penalized problem ).

Proposition

(A priory estimate for penalized solution 1) Under conditions of the proposition ( Existence and uniqueness for evolutionary problem ) the solution $u_{\varepsilon}$ of the problem ( Evolutionary penalized problem ) satisfies the relationship MATH where the $C$ is independent of $\varepsilon$ .

Proof

We act similarly to the proof of the proposition ( Parabolic regularity 1 ). We deduce from the equation of the problem ( Evolutionary penalized problem ): MATH Note that we used the condition MATH on MATH of the proposition ( Existence and uniqueness for evolutionary problem ) so that MATH ). We introduce the following convenience notations MATH so that MATH We have MATH By the condition ( Symmetric principal part ) MATH We substitute MATH , MATH and MATH into $\left( \#\right) $ : MATH We move the terms around MATH and integrate over $\left[ t,T\right] $ . Then, similarly to the proof of the proposition ( Parabolic regularity 1 ), the $B_{0}$ -term estimates from below by ellipticity, all the RHS estimates from above and we obtain the statement of the proposition.





Notation. Index. Contents.


















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