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.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 A. Zero-or-one laws.
 B. Optional random variable. Stopping time.
 C. Recurrence of random walk.
 D. Fine structure of stopping time.
 E. Maximal value of 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.

## Fine structure of stopping time.

roposition

(Wiener-Hopf technique) Let where the functions are Fourier transforms of measures with support on (or ), the functions are Fourier transforms of measures with support on (or respectively).

Suppose that there exists a such that the power series converge for and then

Proof

The equality may be written as We look at terms for every :

According to the proposition ( Inversion of ch.f. into p.m. 1 ), where the is the measure that produces the Fourier transform . There is a similar statement for and . We write the formula in the form and note that the LHS and RHS are produced from measures with disjoint supports. Hence, we must have

With such result in place we proceed to the equation and find and so fourth:

Proposition

(Ch.f. of entrance time 1) Let be the time of first entrance into the interval with being one of the following: , , , :

Then

Proof

We prove the statement for the case .

For an , and being the ch.f. function of we write Using the proposition ( Ch.f. of a sum ) we substitute : and separate the sum into the pre- and post- parts We calculate the first term: We interchange the order of summation : where the is the Fourier transform of the measure having support outside of . Indeed, implies that .

We calculate the second term: We make the change of summation index : By independence of pre- and post- fields, The is distributed like : where the : is the Fourier transform of the measure having support in .

We put our results together: We transform the as follows. The Taylor decomposition of at reads: Therefore, The statement now follows from the proposition ( Wiener-Hopf technique ) and the formula .

Until end of this section, the set and the r.v. are defined as in the proposition ( Ch.f. of entrance time 1 ).

Proposition

We have and iff in which case

Proposition

Suppose that and at least one of is finite, then

Proposition

If and then

Proposition

Suppose that and at least one of is finite. Let , .

1. If but may be then .

2. If then are both finite iff .

 Notation. Index. Contents.