Quantitative Analysis.
Trading Platform.
Python for Excel.
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.
4.Generation of random samples.
5.Monte-Carlo.
6.Convex Analysis.
VI.Basic Math II.
VII.Implementation tools II.
Bibliography.
Forum 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 writeMATH iff $\frac{u}{v}$ is bounded around $x_{0}$ and MATHas $x\rightarrow x_{0}$.

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

3. The expansionMATH are called "asymptotic power series".

Proposition

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 haveMATHMATH andMATH Thus, we must haveMATH ThenMATH 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.

Forum Notation Index Contents


















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