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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
A. Finite difference basics.
B. One dimensional heat equation.
C. Two dimensional heat equation.
D. General techniques for reduction of dimensionality.
E. Time dependent case.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Time dependent case.

uppose the operator of the finite-difference problem MATH is time dependent MATH According to the convergence theorem ( Lax convergence theorem ) we need to prove approximation on the solution $\phi_{t}$ and stability. The stability is proved with the same techniques as in the time independent case. We assume that the spacial operator $A$ has a correct approximation in spacial variables. In this section we concentrate on the problem of $t$ -directional approximation. We assume that the solution $\phi$ possesses smoothness in $t$ -variable and invoke the Taylor decomposition MATH where the higher index refers to the point on the uniform time mesh, the low index is the derivative in $t$ and the $\tau$ is the time step. We repeatedly use the fact that $\phi$ is the solution of $\phi_{t}+A\phi=0$ : MATH MATH MATH MATH MATH With the above results we would like to construct a Crank-Nicolson scheme: MATH for some operator $\Lambda$ that remains to be determined. We expand the above scheme in power of $\tau$ and collect the terms: MATH MATH Hence we need to have MATH and the $\tau$ -term expression gives MATH Hence, the following requirement should be sufficient for approximation in $\tau$ direction MATH Some of the possibilities are MATH MATH MATH

Notation. Index. Contents.

Copyright 2007