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Quantitative Analysis
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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.

Discretization for backward Kolmogorov equation.


e adapt derivations of the section ( Discretization with respect to time parameter ) to backward Kolmogorov equation.

Condition

(Backward Kolmogorov PDE setup) We adopt the setup ( Generic parabolic PDE setup ) with the following exception. We are considering the problem

MATH (Backward Kolmogorov PDE)
for some $u_{T}\in G$ and MATH

The evolution of the solution $u$ happens in opposite direction: from $h$ at time $T$ to time $0$ . We adopt notation and constructs of the sections ( Weak formulation with respect to time parameter ) and ( Discretization with respect to time parameter ). Thus, we introduce the following definitions.

Definition

(Class $\QTR{cal}{D}_{T}$ ) We introduce the class of functions MATH

For any MATH , we apply the operation MATH to the equation ( Backward Kolmogorov PDE ) and integrate over $\left[ 0,T\right] $ : MATH We integrate the first term by parts: MATH

Problem

(Weak backward Kolmogorov with respect to time) In context of the condition ( Backward Kolmogorov PDE setup ) we seek MATH such that

MATH (Time-weak backward Kolmogorov)
for any MATH .

We redefine the partition to reflect backward evolution with time: MATH

Definition of the class MATH remains the same: MATH We introduce the notations MATH for MATH .

We calculate for $v\in S_{\tau}^{q}$ and MATH : MATH Note that MATH We continue MATH

The equation ( Time-weak backward Kolmogorov ) transforms into MATH for any MATH . Thus, MATH . We replace MATH with $w\in S_{\tau}^{q}$ : MATH We merge conditions and arrive to MATH We fix $T_{n}$ and take a $w$ with support within MATH . We arrive to the following problem.

If $n+1<N$ then MATH If $n+1=N$ then MATH

Problem

(Backward discontinuous Galerkin time-discretization) In context of the condition ( Backward Kolmogorov PDE setup ) we seek $v\in S_{\tau}^{q}$ such that for every MATH and any MATH , MATH

Note that the solution of the equation ( Backward Kolmogorov PDE ) satisfies the above problem.





Notation. Index. Contents.


















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