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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.
A. Change of variables for parabolic equation.
B. Discontinuous Galerkin technique.
a. Weak formulation with respect to time parameter.
b. Discretization with respect to time parameter.
c. Discretization for backward Kolmogorov equation.
d. Existence and uniqueness for time-discretized problem.
e. Convergence of discontinuous Galerkin technique. Adaptive time stepping.
C. Laplace quadrature.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Existence and uniqueness for time-discretized problem.


roposition

Under condition ( Generic parabolic PDE setup ) the problem ( Discontinuous Galerkin time-discretization ) has a unique solution on $T_{n}$ for given $f$ and $v_{n-1}$ .

Proof

The class $S_{\tau}^{q}$ is a finite dimensional space. Hence, we only need to prove uniqueness.

The homogenous problem takes the form MATH Assume that $v\in S_{\tau}^{q}$ is a solution of MATH . Set $w=v$ then MATH and MATH Thus MATH All terms in the last relationship are non negative. Therefore MATH





Notation. Index. Contents.


















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