Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
A. Basic properties of characteristic function.
B. Convergence theorems for characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Convergence theorems for characteristic function.


roposition

(Convergence of p.m. and ch.f. 1) Let MATH be a family of p.m. on $\QTR{cal}{R}$ and MATH is the corresponding family of ch.f.

If MATH vaguely then $f_{n}\rightarrow f$ uniformly in every finite interval and MATH is equicontinuous.

Proof

If MATH vaguely then the pointwise convergence of $f_{n}$ follows from the proposition ( Vague convergence as a weak convergence 2 ). To prove the equicontinuity we consider the difference MATH for small $h$ . Since MATH are p.m. the limit $\mu$ is also a p.m. Fix an $\varepsilon>0$ . There exists a positive MATH from the continuity set of $\mu$ such that MATH . By vague convergence there exists an MATH such that $\forall n>n_{0}$ we also have MATH . Therefore, we continue MATH To estimate the first integral we note that the $h$ is a small parameter and $x$ is bounded by $A$ . Hence, it follows from Taylor decomposition of $e^{x}$ around $x=0$ that there is an MATH and a constant $C$ such that MATH and MATH we have MATH . Hence, we continue MATH This proves equicontinuity. Since any ch.f. is bounded MATH the uniform convergence on any finite interval follows from the proposition ( Arzela-Ascoli compactness criterion ).

Proposition

(P.m. vs ch.f. estimate) Let $\mu$ and $f$ be related p.m. and ch.f. Then for any $A>0$ we have MATH

Proof

We use the technique of the proof of the proposition ( Inversion of ch.f. into p.m. 1 ). For this reason the verbose justification of some steps is omitted. MATH Fix a constant $A>0$ . For MATH we have MATH . For MATH we have MATH . We continue MATH Set $T=A^{-1}$ then MATH

Proposition

(Convergence of p.m. and ch.f. 2) Let MATH be a family of p.m. on $\QTR{cal}{R}$ and MATH be the corresponding family of ch.f. Assume

A. MATH pointwise everywhere on $\QTR{cal}{R}$ .

B. $f_{\infty}$ is continuous at $0$ .

Then we have

a. MATH vaguely and $\mu_{\infty}$ is a p.m.

b. $f_{\infty}$ is a ch.f. of $\mu_{\infty}$ .

Proof

According to the proposition ( Vague precompactness of s.p.m. ), there is a subsequence $\mu_{n_{k}}$ that converges vaguely to some s.p.m. $\mu$ . We proceed to show that $\mu$ is a p.m.

Fix a large constant $A$ such that $\pm2A$ are continuity points of $\mu$ . We have MATH By the proposition ( Equivalent definitions of vague convergence ) MATH We seek to apply the proposition ( P.m. vs ch.f. estimate ) to the above expression. Set $h=A^{-1}$ . We estimate MATH Fix $\varepsilon>0$ . It follows from MATH and continuity of $f_{\infty}$ at $0$ that MATH such that $\forall h<h_{0}$ MATH With MATH so fixed we use the convergence of $f_{n}$ to $f_{\infty}$ , boundedness MATH and the proposition ( Dominated convergence theorem ) to state that MATH such that $\forall n>N$ MATH Hence, MATH We apply the proposition ( P.m. vs ch.f. estimate ): MATH Therefore MATH for arbitrary $\varepsilon$ . We proved that the $\mu$ is a p.m.

Let $f$ is ch.f. of $\mu$ . Then, by assumption (A) and the proposition ( Convergence of p.m. and ch.f. 1 ), $\ f=f_{\infty}$ . Therefore, every limit point $\mu$ considered above has the same ch.f. Hence, by the proposition ( Inversion of ch.f. into p.m. 1 ) every limit point of MATH is the same. Hence, $\mu_{n}$ converges vaguely into a p.m.





Notation. Index. Contents.


















Copyright 2007