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.
B. One dimensional heat equation.
a. Finite difference schemes for heat equation.
b. Stability of one-dim heat equation schemes.
c. Remark on stability of financial problems.
d. Lagrangian coordinate technique.
e. Factorization procedure for 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.

Finite difference schemes for heat equation.

onsider the following boundary problem: MATH MATH MATH MATH where the MATH is the unknown function, the functions MATH are given and regular, and the variable $x$ and $t$ lie in the domain MATH We set up the lattice MATH and approximate MATH with the $\Lambda_{x}u$ . We arrive to the following ODE problem MATH MATH MATH MATH for $k=1,2,...,N-1$ , where MATH , MATH , MATH .

Let us consider the boundary conditions. The matrix of the Laplacian $\Lambda$ has the form MATH Note what happens to the finite difference approximation of the second derivative on the edges of the matrix. Obviously, we do not approximate it if MATH MATH are some non zero values. We perform the following trick. We set MATH Then the equation MATH will describe exactly the same MATH if we choose $g_{k}$ according to

MATH (Boundary trick)
MATH MATH Set up the lattice MATH covering the $D_{t}$ and integrate the $k$ -th equation over the interval MATH . We have MATH where MATH MATH MATH The integral may be approximated by one of the quadrature formulas MATH MATH MATH The resulting schemes are called implicit, explicit and Crank-Nicolson schemes respectively. The expressions for the schemes are
MATH (Implicit scheme)
MATH (Explicit scheme)
MATH (Krank Nicolson)
with the boundary conditions MATH in every case.

Notation. Index. Contents.

Copyright 2007