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.

## Random walk.

e consider the sequence of r.v. and a triple . The is the event space of and is the minimal -algebra that makes all measurable. We think of as a consecutive sequence of trials as increases. Hence, is a "process" and is a time parameter.

We use the notation to represent the -algebra containing the information available up to time . Hence, is the minimal -algebra that makes the family measurable.

We use the notation to represent the -algebra containing the information that comes after the time . Hence, is the minimal -algebra that makes the the family measurable.

We use the notation to represent the minimal -algebra that contains all of the , . By the minimality of we have

Proposition

(Random walk space approximation). Given and there exists such that

We use the notation for the intersection

 (Remote field)

Sometimes it is convenient to think of as the product space

 (Random walk space)
where each is the probability space for . The is the product measure consistent with the d.f. on each . The is a collection of infinite sequences of real numbers: The denotes the "shift":
 (Shift)
The is the applied times.

The denotes the set of permutations of integers from the range . A permutation (and similarly ) produces a mapping on and on according to the rules

 (Permutation)

Definition

(Invariant set) The set is called "invariant" if .

Definition

(Permutable set) The set is called "permutable" if for every permutation of finite number of positions. A function is "permutable" if .

Definition

(Remote event). Any set (see the formula ( Remote field )) is called "remote event".

Clearly, an invariant set is remote and a remote set is permutable.

Definition

(Independent process) The family is an "independent" process if are independent r.v. The family is "stationary independent" process if are iid.

Definition

(Random walk) For a stationary independent process we defined the "random walk" as the process , where and for .

 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.
 Notation. Index. Contents.