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 where the is some differential operator acting on some spacial variables. Finite difference scheme for such equation regularly has the form

 (Evolution equation)
where the is some spacial finite difference operator with complete system of eigenvectors for every . Hence, we start from some initial condition and use the evolution equation ( Evolution equation ) to recursively evaluate for all times. Let is the spectrum of . The condition 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 is a bounded linear operator in a Banach space and the function is analytic in a neighborhood of . Then

 (Spectral mapping)

Proposition

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

 (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 there exists matrixes and such that and

Proposition

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

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 be any square matrix. There exist matrixes and with the following properties. and has the following block-diagonal form

We conclude from the above theorem that Consequently, the matrix is normal if and only if all 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 . In general,

 Notation. Index. Contents.