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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.

Existence and uniqueness for evolutionary problem.


roposition

(Existence and uniqueness for evolutionary problem) Suppose that the bilinear form is defined by ( Bilinear form B 2 ), the condition ( Symmetric principal part ) holds and MATH MATH MATH MATH Then there exists a unique solution MATH of the problem ( Evolutionary variational inequality problem ) and MATH

Proof

(Uniqueness) Suppose existence of two solutions $u_{1}$ and $u_{2}$ of the problem ( Evolutionary variational inequality problem ): MATH

We set $u=u_{1}$ and $v=u_{2}$ and then we also set $u=u_{2}$ and $v=u_{1}$ : MATH and add: MATH Let $w=u_{1}-u_{2}$ then we have MATH According to the proposition ( Energy estimates for the bilinear form B )-1: MATH thus MATH We integrate over $\left[ t,T\right] $ and use MATH to deduce MATH We drop the positive term MATH : MATH Assume that MATH . Let $t^{\ast}$ be the rightmost local maximum of MATH on $\left[ 0,T\right] $ so that MATH is decreasing on MATH and MATH . Then application of the mean value theorem leads to MATH which leads to contradiction with the decreasing behavior of MATH as we put $h\downarrow0$ . Therefore MATH .





Notation. Index. Contents.


















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