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Quantitative Analysis
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Numerical Analysis
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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.

Definitions and main convergence theorem.


uppose $D\,$ is a closed domain in $R^{N}$ . Let MATH be a lattice with step $h$ covering $D$ . We introduce functions $u$ with domain in $D_{h}$ and range in $R^{M}$ . Finite difference operator $A^{h}$ is operator acting on such functions. The regular notation is MATH . We write MATH and denote MATH the restriction of a regular function $v$ on $D_{h}.$ Sometimes we also use notation $u^{h}$ for a function defined on $D_{h}$ .

For example, MATH

The finite difference operator $A_{h}$ approximates $A$ on the function $u$ with order $n$ if there exist constants $H,M$ such that for any $h<H$ MATH Similarly, one defines approximation of a function $u$ by a function $u^{h}$ by means of the inequality MATH

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

The finite difference problem ("scheme") MATH is stable if there exist constants $H,C$ such that for any $h<H$ the solution $u^{h}$ satisfies MATH 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 $A$ be the differential operator of the linear boundary problem $Au=f$ on $D$ and $A^{h}u^{h}=f^{h}$ be the linear finite difference scheme on $D_{h}$ covering $D$ . Assume that the following conditions are met:

1. $A^{h}$ approximates $A$ on the solution $u$ of the problem $Au=f$ with order $n$ .

2. $f^{h}$ approximates $f$ with order $n$ .

3. The finite difference scheme is stable.

Then the solution $u^{h}$ of the scheme converges to the solution $u$ of the problem with order $n$ as $h\rightarrow0$ and the following estimate holds MATH where the $C$ is the constant from the definitions of stability and the $M_{A},M_{f}$ are the approximation constants for $A$ and $f$ .

Proof

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





Notation. Index. Contents.


















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