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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
A. Asymptotic expansion of Laplace integral.
B. Asymptotic expansion of integral with Gaussian kernel.
C. Asymptotic expansion of generic Laplace integral. Laplace change of variables.
D. Asymptotic expansion for Black equation.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Asymptotic expansions.


symptotic expansions are useful for obtaining high performance analytical formulas, for validation of solutions, acceleration of Monte-Carlo convergence, obtaining preconditioners and adaptive grids. The references for this section are [Erdelyi] and [Sveshnikov] . The presentation here is partial.

Definition

(O symbols) Let $X$ be a Banach space.

1. For functions MATH we write MATH iff $\frac{u}{v}$ is bounded around $x_{0}$ and MATH as $x\rightarrow x_{0}$ .

2. The series MATH is "asymptotic expansion" of MATH at $x_{0}$ iff MATH We will use notation MATH

3. The expansion MATH are called "asymptotic power series".

Proposition

(Differentiation of asymptotic power series) If MATH is differentiable and both $f$ and $f^{\prime}$ possess asymptotic power series then such power series are connected by termwise differentiation.

Proof

The proof is based on assertion that there is similar result for integration, see the proposition ( Dominated convergence theorem ) or less generic results from calculus. We have MATH MATH and MATH Thus, we must have MATH Then MATH and we match terms with the expansion for $f$ .




A. Asymptotic expansion of Laplace integral.
B. Asymptotic expansion of integral with Gaussian kernel.
C. Asymptotic expansion of generic Laplace integral. Laplace change of variables.
D. Asymptotic expansion for Black equation.

Notation. Index. Contents.


















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