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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.

Convergence of discontinuous Galerkin technique. Adaptive time stepping.


roposition

(Convergence of discontinuous Galerkin technique) Assume that the condition ( Generic parabolic PDE setup ) takes place and $u$ is a solution of the problem ( Generic parabolic PDE problem ). Let $v$ be a solution of the problem ( Discontinuous Galerkin time-discretization ) applied consecutively to the time intervals MATH . Then MATH

Proof

We introduce a mapping MATH with the following properties: MATH thus, for each MATH the $\tilde{u}$ equals $u$ at $t_{n}$ but does not equal $u$ at $t_{n-1}$ . We proceed to prove existence and uniqueness of such $\tilde{u}$ . Let MATH be an orthonormal basis of $G$ and MATH for some sequence MATH . For every fixed $m$ and $n$ we need MATH to satisfy MATH Thus, for each fixed $m,n$ we have $q$ equations for $q$ unknowns MATH . To complete the proof of existence of MATH it remains to show that the homogenous system of equations MATH has only the trivial solution. Let MATH then, by $\left( \#\right) $ , MATH for some MATH and, by MATH , MATH but also MATH and both the terms MATH and MATH do not change sign on $T_{n}$ . Thus, we must have MATH on $T_{n}$ . Existence of $\tilde{u}$ has been established.

It follows from such result that if $u\left( t\right) $ is a $t$ -polynomial of degree $q-1$ then MATH would approximate it exactly.

According to the proposition ( Integral form of Taylor decomposition ), for a function MATH we have MATH where MATH . By polynomial approximation up to degree $q-1$ , we must have MATH Indeed, if one of the derivatives MATH is not zero then we cannot have the polynomial approximation with the same formula. Thus MATH where MATH or MATH Consequently MATH We use the formula ( Cauchy inequality ), MATH Then, by orthogonality of MATH , MATH and by properties of the operator $A$ listed in the condition ( Generic parabolic PDE setup ), MATH

For $u$ being a solution of the problem ( Generic parabolic PDE problem ), $v$ be a solution of the problem ( Discontinuous Galerkin time-discretization ) and $\tilde{u}$ being the polynomial approximation defined in $\left( \&\right) $ we decompose the error MATH We already have the desired estimate for $\rho=\tilde{u}-u$ . It remains to estimate $\theta=v-\tilde{u}$ . Note that both $v$ and $u$ are solutions of the equation of the problem ( Discontinuous Galerkin time-discretization ), hence, for $h\equiv v-u$ we have MATH Consequently MATH We calculate the part MATH By definition of $\tilde{u}$ there is orthogonality of $\rho\,\ $ to $t$ -variable polynomials up to degree $q-2$ . Thus, we perform integration by parts. MATH Now the integral is zero by the orthogonality because $w\in S_{\tau}^{q}$ thus MATH is a $\left( q-2\right) $ -degree $t$ -variable polynomial. MATH The $\rho$ is zero at MATH by definition of $\tilde{u}$ . MATH

Therefore, we arrived to MATH We set $w=\theta$ and evaluate the part MATH We use the formula ( Cauchy inequality ). MATH We arrived to MATH or MATH Hence MATH Thus MATH MATH Note that $\theta_{0}=0$ by definition of the components and MATH . MATH and we use the estimate $\left( \$\right) $ MATH





Notation. Index. Contents.


















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