8-Stochastic Integrals Page 232 Wednesday, February 4, 2004 12:50 PM
232 The Mathematics of Financial Modeling and Investment Managementinstance, weekly or monthly time series look alike. Recent empirical and
theoretical research work has made this claim more precise as we will
see in Chapter 13.
Let’s consider a one-dimensional standard Brownian motion. If we
wait a sufficiently long period of time, every path except a set of paths
of measure zero will return to the origin. The path between two consec-
utive passages through zero is called an excursion of the Brownian
motion. The distribution of the maximum height attained by an excur-
sion and of the time between two passages through zero or through any
level have interesting properties. The distribution of the time between
two passages through zero has infinite mean. This is at the origin of the
so-called St. Petersburg paradox described by the Swiss mathematician
Bernoulli. The paradox consists of the following. Suppose a player bets
increasing sums on a game which can be considered a realization of a
random walk. As the return to zero of a random walk is a sure event,
the player is certain to win—but while the probability of winning is one,
the average time before winning is infinite. To stay the game, the capital
required is also infinite. Difficult to imagine a banker ready to put up
the money to back the player.
The distribution of the time to the first passage through zero of a
Brownian motion is not Gaussian. In fact, the probability of a very long
waiting time before the first return to zero is much higher than in a nor-
mal distribution. It is a fat-tailed distribution in the sense that it has
more weight in the tail regions than a normal distribution. The distribu-
tion of the time to the first passage through zero of a Brownian motion
is an example of how fat-tailed distributions can be generated from
Gaussian variables. We will come back on this subject in Chapter 13
where we deal with the question of how the fat-tailed distributions
observed in financial markets are generated from a large number of
apparently independent events.STOCHASTIC INTEGRALS DEFINED
Let’s now go back to the definition of stochastic integrals, starting with
one-dimensional stochastic integrals. Suppose that a probability space
(Ω,ℑ,P) equipped with a filtration ℑt is given. Suppose also that a
Brownian motion Bt(ω) adapted to the filtration ℑt is given. We will
define Itô integrals following the three-step procedure outlined earlier in
this chapter. We have just completed the first step defining Brownian
motion. The second step consists in defining the Itô integral for elemen-
tary functions.