5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 163
Paris stock exchange, and so is the originator of the idea of what is now
called Brownian motion. This term is occasionally found in financial
literature and European usage.Mathematical Ideas
5.4 Transformations of the Wiener Process
Previously, we have considered adiscrete time random process, that is, at
timesn= 1, 2 , 3 ,...corresponding to coin flips, we have a sequence of random
variablesTn. We are now going to consider acontinuous time random process,
that is a functionWtwhich is a random variable at each timet≥0. To say
Wtis a random variable at each time is too general so we must put some
additional restrictions on our process to have something interesting to study.
Definition(Wiener Process).TheStandard Wiener Processis a stochas-
tic processW(t), fort≥0, with the following properties:
- Every incrementW(t)−W(s) over an interval of lengtht−sis normally
distributed with mean 0 and variancet−s, that is
W(t)−W(s)∼N(0,t−s)- For every pair of disjoint time intervals [t 1 ,t 2 ] and [t 3 ,t 4 ], witht 1 <
t 2 ≤ t 3 < t 4 , the incrementsW(t 4 )−W(t 3 ) andW(t 2 )−W(t 1 ) are
independent random variables with distributions given as in part 1, and
similarly forndisjoint time intervals wherenis an arbitrary positive
integer. - W(0) = 0
- W(t) is continuous for allt.
Note that property 2 says that if we knowW(s) =x 0 , then the indepen-
dence (andW(0) = 0) tells us that no further knowledge of the values of
W(τ) forτ < shas any additional effect on our knowledge of the probability
law governingW(t)−W(s) witht > s. More formally, this says that if
0 ≤t 0 < t 1 < ... < tn< t, then
P[W(t)≥x|W(t 0 ) =x 0 ,W(t 1 ) =x 1 ,...W(tn) =xn] =P[W(t)≥x|W(tn) =xn]