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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.
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.

Existence and uniqueness for non-coercive stationary problem.


ondition

(Non-coercivity assumption 1) The coefficient $c\left( x\right) $ of the formula ( Operator L ) satisfies MATH for some constant $c_{0}$ .

Remark

(Preliminary reduction) It does not restrict generality to assume that MATH for a constant $f_{0}$ .

Indeed, since $U$ is bounded there exists a constant $c$ such that MATH then we set $\phi$ to be a solution of the problem MATH and consider $\tilde{u}=u+\phi$ instead of $u$ .

Proposition

(Existence and uniqueness for non-coercive stationary problem.) Suppose that the condition ( Non-coercivity assumption 1 ) holds and the functions $f,\psi$ of the problem ( Stationary variational inequality problem ) satisfy MATH Then there exists a unique solution $u$ of the problem ( Stationary variational inequality problem ) and MATH

Proof

(Existence) Let MATH is such that MATH for some $\tilde{f}$ such that $\tilde{f}\geq f$ on $U$ . For example, given the condition ( Non-coercivity assumption 1 ), $u_{0}$ can be a large enough constant so that MATH .

According to the proposition ( Energy estimates for the bilinear form B ) there exist constants $\lambda>0,\alpha>0$ such that MATH

According to the proposition ( Existence and uniqueness for stationary problem ) there is a sequence MATH such that MATH

We now prove that $u_{0}\geq u_{1}\,$ a.e. in $U$ . Indeed, the $u_{0},u_{1}$ are given by the relationships MATH Note that we cannot apply the proposition ( Monotonicity of solution of stationary problem ) because the operators of the problems are different and because $\tilde {f}\geq f$ but $\lambda u_{0}\geq0$ . However, we act similarly to the proof of the proposition ( Monotonicity of solution of stationary problem ). We set MATH in the first relationship and MATH in the second: MATH add: MATH and move terms MATH Note that MATH . Hence MATH Therefore, we deduce MATH or MATH where MATH Hence, by the choice of $\lambda$ in MATH , MATH

Next, we prove that $u_{n}\geq u_{n+1}$ a.e. for $n=1,2,...$ Indeed, we use the recursion and assume that the same is proven for $n-1$ . The $u_{n}\,$ and $u_{n-1}$ are given by the relationships MATH Hence, by the assumption of recursion, the proposition ( Monotonicity of solution of stationary problem ) applies.

Next, we show that $u_{n}\geq-C$ for some constant $C>0$ using the recurrence argument.

We choose $C$ so that MATH We set MATH in the equation $\left( \&\right) $ so that MATH Note that MATH hence MATH MATH We use MATH : MATH MATH and apply the MATH : MATH By the recurrence assumption we have $u_{n-1}+C\geq0$ and by the choice of $C$ we have MATH . Hence MATH . Therefore, MATH

We collect the previous results into the relationship MATH

We now complete the proof of existence as follows. We fix $v$ in $\left( \&\right) $ and rewrite it using MATH MATH MATH Combining it with $\left( \#\right) $ and boundedness of $U$ we deduce MATH

Therefore, according to the proposition ( Weak compactness of bounded set ), MATH then by the proposition ( Rellich-Kondrachov compactness theorem ), $\left( \#\right) $ and boundedness of $U$ MATH and MATH We now pass the relationship $\left( \&\right) $ : MATH to the limit $n\rightarrow\infty$ : MATH thus MATH





Notation. Index. Contents.


















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