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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.
a. Peaceman-Rachford (alternating directions) scheme.
b. Stability of Peaceman-Rachford.
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.

Stability of Peaceman-Rachford.

e aim to convert the scheme ( Alternating directions1 ),( Alternating directions2 ) to the canonical form by eliminating the $\tilde{u}$ . Add the equations ( Alternating directions1 ),( Alternating directions2 ): MATH and substitute $\tilde{u}$ from ( Alternating boundary ): MATH Note that $\Lambda_{x}$ and $\Lambda_{y}$ are commutative. We transform the last expression as follows: MATH MATH MATH Finally, we write the evolution equation MATH where MATH MATH MATH For stability it is sufficient to have MATH Such results follow from the spectral considerations of the Crank-Nicolson ( Crank Nicolson spectrum ) and implicit ( Implicit spectrum ) schemes and the minimax theorem ( Minmax theorem ).

One may ask "Wait a moment, how about the boundary conditions?". The answer is "Check the trick around the ( Boundary trick )".

Notation. Index. Contents.

Copyright 2007