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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.
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.

Stability of general evolution equation.


or purposes of financial problems we are always interested in equations of the form MATH where the $L$ is some differential operator acting on some spacial variables. Finite difference scheme for such equation regularly has the form

MATH (Evolution equation)
where the $H_{t}$ is some spacial finite difference operator with complete system of eigenvectors for every $t$ . Hence, we start from some initial condition $u_{0}$ and use the evolution equation ( Evolution equation ) to recursively evaluate $u$ for all times. Let MATH is the spectrum of $H_{t}$ . The condition MATH is a sufficient condition for stability of the scheme in such a situation.

Two more statements are directly relevant to the subject of stability.

Proposition

(Spectral mapping theorem). Suppose $H$ is a bounded linear operator in a Banach space and the function $f\left( x\right) $ is analytic in a neighborhood of MATH . Then

MATH (Spectral mapping)

Proposition

(Minimax theorem). Suppose $H$ is a bounded normal ( MATH ) operator in a Hilbert space. Then

MATH (Minmax theorem)

The spectral minimax theorem is a consequence of the following decomposition statement.

Proposition

(Spectral representation of normal matrix). For every square normal matrix $A$ there exists matrixes $Q$ and $\Lambda$ such that MATH and MATH

Proposition

(Gershgorin circle theorem). Let MATH then MATH and MATH where the MATH is the ball on the complex plane centered at $a$ with the radius $b$ .

Observe that the above statement provide a complete recipe for determining norm of a normal matrix and thus establishing stability of an evolution scheme with a normal operator. Naturally, we would like to have some extension to the situation without normality. The following considerations show that for most practical purposes there is no such extension. Consequently, one uses energy considerations and Galerkin (finite element) technique (see the chapter ( Finite elements chapter )) if the normality cannot be preserved.

Proposition

(Jordan decomposition theorem). Let MATH be any square matrix. There exist matrixes $B$ and $H$ with the following properties. MATH and $H$ has the following block-diagonal form MATH MATH

We conclude from the above theorem that MATH Consequently, the matrix is normal if and only if all MATH are equal to 1.

To see that the spectral analysis does not deliver simple stability results in the situation without normality let us evaluate the norm of the matrix MATH . MATH In general, MATH





Notation. Index. Contents.


















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