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 Then there exists a unique solution of the problem ( Evolutionary variational inequality problem ) and

Proof

(Uniqueness) Suppose existence of two solutions and of the problem ( Evolutionary variational inequality problem ):

We set and and then we also set and : and add: Let then we have According to the proposition ( Energy estimates for the bilinear form B )-1: thus We integrate over and use to deduce We drop the positive term : Assume that . Let be the rightmost local maximum of on so that is decreasing on and . Then application of the mean value theorem leads to which leads to contradiction with the decreasing behavior of as we put . Therefore .

 Notation. Index. Contents.