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.

## Maximal value of random walk.

e will be using the notation of the proposition ( Maximum of random walk ).

Notation

For a random walk we introduce the following quantities: The is understood as a permutation acting on .

Proposition

(Spitzer identity)

1. We have for :

2. is finite a.s. iff in which case we have

Proof

(1) We calculate Note that . Indeed, the says that among positions the maximum is attained at -th and the says that among positions the maximum is attained at . Hence, we continue Note that and belong to pre- and post- event fields, hence, these are independent: The is distributed as : We write the expression of interest and substitute the latest result: The last expression has the structure . One may see by concentrating at each term that such structure is the product . Hence, We calculate each term of the formula . Note that . Also, note that . Hence, . We continue: At this point we invoke the proposition ( Ch.f. of entrance time 1 ): with substituted for and and obtain We calculate the second term of the formula similarly, using : We collect the results:

Proposition

Notation

We introduce the r.v. and as follows

Proposition

For each the random vectors and have the same distribution.

For each the random vectors and have the same distribution.

Proposition

For

Proposition

If the common distribution of stationary independent process is symmetric then

 Notation. Index. Contents.