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.

## Basic properties of characteristic function.

roposition

(Uniform continuity of ch.f.) Characteristic function of a r.v. is uniformly continuous in .

Proposition

(Ch.f. of a sum) If and are independent r.v. with d.f. and then the r.v. has a d.f. and ch.f. .

Proof

For the part use the formula ( Total_probability_rule ). The second part is a direct verification.

Proposition

(Inversion of ch.f. into p.m. 1) If is a p.m. induced by a r.v. then for we have

Proof

We proceed by direct verification. We substitute the definition of into the claim of the proposition: We would like to invert the order of integration using the proposition ( Fubini theorem ). Hence, we need to verify that the function is integrable over . We estimate and the integral is finite. Hence, the proposition ( Fubini theorem ) is applicable and we reverse the order of integration: We calculate the integral with respect to : We expand the last integral with the help of and note that is an odd function of . Hence, . where the last equality holds because is an even function of . The above is being passed to the limit . Hence, the next task is to calculate the integral of the form First, we calculate the integral The function is sharply decaying a positive infinities for and . Hence, the proposition ( Fubini theorem ) applies and we reverse the order of integration: We calculate the internal integral via the repeated integration by parts: Then Note that if is positive then If is negative then If is zero then Hence,

We substitute the last result into the integral Hence, by the proposition ( Dominated convergence theorem ) the limit interchanges with the integral and we recover the result:

Proposition

(Inversion of ch.f. into p.m. 2) If is a p.m. induced by a r.v. then

Proposition

(Inversion of chf into df). Let be a r.v. with ch.f. and d.f. . Assume that . Then is continuously differentiable and

 Notation. Index. Contents.