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.

## Definitions and main convergence theorem.

uppose is a closed domain in . Let be a lattice with step covering . We introduce functions with domain in and range in . Finite difference operator is operator acting on such functions. The regular notation is . We write and denote the restriction of a regular function on Sometimes we also use notation for a function defined on .

For example,

The finite difference operator approximates on the function with order if there exist constants such that for any Similarly, one defines approximation of a function by a function by means of the inequality

For example, the defined above finite difference operator approximates with order 2 on any smooth function.

The finite difference problem ("scheme") is stable if there exist constants such that for any the solution satisfies The following theorem (called "Lax convergence theorem") states that if the scheme approximates and is stable then it converges to the right solution.

Theorem

Let be the differential operator of the linear boundary problem on and be the linear finite difference scheme on covering . Assume that the following conditions are met:

1. approximates on the solution of the problem with order .

2. approximates with order .

3. The finite difference scheme is stable.

Then the solution of the scheme converges to the solution of the problem with order as and the following estimate holds where the is the constant from the definitions of stability and the are the approximation constants for and .

Proof

The satisfies the problem Hence, by stability, and by approximation,

 Notation. Index. Contents.