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.

## Recurrence of random walk.

ccording to the proposition ( Infinitely often zero-or-one law ), for a random walk

Definition

(Recurrent value of random walk) The number is called "recurrent value" of random walk if

The set of recurrent values is denoted by .

Definition

(Possible value of random walk) The number is called "possible value" of random walk if

Proposition

(Structure of recurrent values) The set is described by one of the following statements.

1. .

2.

3. .

4. .

Proof

Let . Then for any and almost any path Hence, The r.v. is distributed like . Therefore, . We proved that the set is an additive group.

One may show by a similar argument that the set is closed.

Let . The cannot have a bounded neighborhood around it that is also included in . Indeed, by additivity of one then expands such neighborhood beyond any boundary. Hence, either is or it consists of singular points. In the latter case is either or has the form due to closeness with respect to addition.

Proposition

(Extension of Borel-Cantelli lemma to random walk). For a random walk ,

1. .

2. .

Note that means and means .

Proof

The statement 1 is a direct consequence of the proposition ( Borel-Cantelli lemma, part 1 ). The statement 2 does not follow from ( Borel-Cantelli lemma, part 2 ) because are not independent events. Hence, we proceed with the proof of 2.

By the definition ( Limsup and liminf for sets ), for a family of sets then Thus, for any , According to the section ( Operations on sets and logical statements ), the statement means . Therefore, we continue The above statements within the union are disjoint: Note that and imply . Hence, and the events and are independent: Note that is distributed as : . Hence, the second probability is -independent: If follows that if then we must have .

Next, we complete the proof by showing that for any . We introduce the event and note that the events are disjoint. Hence, we have and, consequently, We now repeat the argument that lead to starting from .

Proposition

(Recurrence lemma 1) For a random walk , any and any we have

Proof

We calculate We define and include the event in the union : The term joins the as the -th term: We change the order of summation: The and sums are separate:

Proposition

(Recurrence lemma 2) Let be a family of positive numbers such that

1. is increasing in and as .

2. .

3. as .

Then

Proof

Assume the contrary. We calculate by 2: by 1: We pass the inequality to the limit and apply 3 with then for arbitrary . This is a contradiction.

Proposition

(Recurrence result 1) If in pr. then

Proof

We apply the proposition ( Recurrence lemma 2 ) with . The condition yields the condition 3 of the proposition ( Recurrence lemma 2 ) and the proposition ( Recurrence lemma 1 ) yields the condition 2 of the proposition ( Recurrence lemma 2 ). We derive Thus according to the second part of the proposition ( Extension of Borel-Cantelli lemma to random walk ) applied to .

Proposition

(Recurrence result 2) Suppose that at least one of or is finite. Then iff . Otherwise iff and iff .

Proof

The statement is a consequence of the propositions ( Strong law of large numbers for iid r.v. ) and ( Recurrence result 1 ).

 Notation. Index. Contents.