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 be a family of p.m. on and is the corresponding family of ch.f.

If vaguely then uniformly in every finite interval and is equicontinuous.

Proof

If vaguely then the pointwise convergence of follows from the proposition ( Vague convergence as a weak convergence 2 ). To prove the equicontinuity we consider the difference for small . Since are p.m. the limit is also a p.m. Fix an . There exists a positive from the continuity set of such that . By vague convergence there exists an such that we also have . Therefore, we continue To estimate the first integral we note that the is a small parameter and is bounded by . Hence, it follows from Taylor decomposition of around that there is an and a constant such that and we have . Hence, we continue This proves equicontinuity. Since any ch.f. is bounded the uniform convergence on any finite interval follows from the proposition ( Arzela-Ascoli compactness criterion ).

Proposition

(P.m. vs ch.f. estimate) Let and be related p.m. and ch.f. Then for any we have

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. Fix a constant . For we have . For we have . We continue Set then

Proposition

(Convergence of p.m. and ch.f. 2) Let be a family of p.m. on and be the corresponding family of ch.f. Assume

A. pointwise everywhere on .

B. is continuous at .

Then we have

a. vaguely and is a p.m.

b. is a ch.f. of .

Proof

According to the proposition ( Vague precompactness of s.p.m. ), there is a subsequence that converges vaguely to some s.p.m. . We proceed to show that is a p.m.

Fix a large constant such that are continuity points of . We have By the proposition ( Equivalent definitions of vague convergence ) We seek to apply the proposition ( P.m. vs ch.f. estimate ) to the above expression. Set . We estimate Fix . It follows from and continuity of at that such that With so fixed we use the convergence of to , boundedness and the proposition ( Dominated convergence theorem ) to state that such that Hence, We apply the proposition ( P.m. vs ch.f. estimate ): Therefore for arbitrary . We proved that the is a p.m.

Let is ch.f. of . Then, by assumption (A) and the proposition ( Convergence of p.m. and ch.f. 1 ), . Therefore, every limit point considered above has the same ch.f. Hence, by the proposition ( Inversion of ch.f. into p.m. 1 ) every limit point of is the same. Hence, converges vaguely into a p.m.

 Notation. Index. Contents.