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Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

If q __>__ p, conclude from the preceding
problem: In a random walk starting at the origin, the number
of visits to the point a > 0,
that take place before the first return to the geometric
distribution with ratio 1 - qq_{a-1}.
(Why is the condition q __>__
p necessary?)

This problem is adapted from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter XIV, Section 9, problem 3, page 367.

This problem is conceptually and computationally simpler if p = 1/2 = q. It is convenient to first calculate and record some probabilities, then solve the problem.

Starting from the origin, the probability to reach a > 0 before returning to the
origin is p(1 - q_{1}) =
p(1 - (1 - 1/a))
= p(1/a) = 1/(2a).

Starting at a, the probability
to reach the origin before returning to the starting point is
qq_{a-1} = q(1
- (a -
1)/a) = q(1/a) = 1/(2a).
This makes sense, since this situation is symmetric with the
previous situation and so should have the same probability.

The ratio for the geometric probability is supposed to be 1
- qq_{a-1} = 1
- q(1 -
(a - 1)/a)
= 1 - q(1/a) = 1
- 1/(2a) = (2a - 1)/(2a).

The probability of 0 visits to a before hitting the origin is
the complement of the probability of one (or more) visits to
a before hitting the origin. Since the random walk is
recurrent (that is, will hit every point eventually from any
starting point, this is where the necessity of p = q = 1/2 comes in!) the probability of
hitting the origin again from a
(perhaps after more visits to a)
is certain, so the probability of one (or more) visits to
a before hitting the origin is
the same as the probability of a first visit to a, namely p(1
- q_{1}) =
1/(2a). Then the desired probability of 0 visits
is (1 - 1/(2a)) =
(2a - 1)/(2a).

The probability of exactly one visit to the origin is the probability of passing from the origin to a without first returning to the origin, followed by passing from a to the origin without first returning to a. This is

The probability of exactly two visits to a before returning to the origin is the probability of passing from the origin to a without first returning to the origin, followed by a bridge from to a to a without touching the origin, followed by passing from a to the origin without first returning to a. The middle probability is the same as the already computed probability of passing exactly 0 visits to a from the origin, by the symmetry of the situation. This is

Now the pattern is clear, and we see that the number of visits to a with out first returning to the origin is a geometric random variable with ratio (2a - 1)/(2a).

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