Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
A. Finite difference basics.
a. Definitions and main convergence theorem.
b. Approximations of basic operators.
c. Stability of general evolution equation.
d. Spectral analysis of finite difference Laplacian.
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.

Spectral analysis of finite difference Laplacian.

n this section we compute the spectrum of the Laplacian $\Lambda$ explicitly. It will be seen that the Gershgorin theorem gives a good boundary.

We introduce the lattice MATH and consider the spectral problem

MATH (Laplace problem)
MATH (Laplace bounds)
We introduce the lattice functions $u^{k}$ with the values MATH Such functions satisfy the boundary conditions ( Laplace bounds ) and constitute a basis in linear algebra sense. We compute action of the operator $\Lambda_{x}$ on $u^{k}$ : MATH MATH MATH MATH MATH MATH MATH Since MATH we have MATH MATH MATH Therefore, $u^{k}$ are eigenvalues and $\lambda_{k}$ constitute the spectrum of $\Lambda$ :
MATH (Laplacian Spectrum)
We have the spectral limits
MATH (Laplacian Limits)

Notation. Index. Contents.

Copyright 2007