3.3 Brownian Motion
報告者：陳政岳
3.3.1 Definition of Brownian Motion
Definition 3.3.1
Let  , F ,   be a probability space. For each w   ,
suppose there is a continuous function W  t  of t  0
that satisfies W  0   0 and that depends on w .
Then W  t  , t  0 , is a Brownian motion if for all
0  t0  t1  …  tm the increments
W  t1   W  t1   W  t0  , W  t 2   W  t1  ,
,W  t m   W  t m 1 
are independent and each of these increments is
normally distributed with
.  W  ti 1   W  ti    0, Var W  ti 1   W  ti    ti 1  ti




3.3.1 Definition of Brownian Motion
Difference Brownian motion W  t  and scaled random
walk W n  t  ：
• Brownian motion：
1. Time steps are not linear.
2. Brownian motion is normal.
• scaled random walk：
1. Time steps are linear.
2. Brownian motion is only approximately normal.
3.3.2 Distribution of Brownian
Motion
A. The increments W  t1   W  t1   W  t0  , W  t2   W  t1  ,
, W  tm   W  t m 1  are independent and normally
distributed, the random variables W  t1  , W  t 2  , ,W  t m 
are jointly normally distributed. The joint distribution
of jointly normal random variables is determined by
their means and covariances.
3.3.2 Distribution of Brownian
Motion
1. W  ti  has mean zero.
2. the covariance of W  s  and W  t ：
 W  s  W  t     W  s  W  t   W  s    W 2  s  
= W  s    W  t   W  s   + W 2  s  
=0+Var W  s    s
3.3.2 Distribution of Brownian
Motion
B. The moment-generating function of this
random vector can be computed using the
moment-generating function for a zero-mean
normal random variable with variance t and
the independence of the increments.
Ps. the moment-generating function
f  x 
1
2 t
e
x2

2t
, the moment   u   e
1 2
u t
2
3.3.2 Distribution of Brownian
Motion
umW  tm   um 1W  tm 1   u m  2W  t m  2  
 u1  t1 
 um W  tm   W  tm 1     um 1  um  W  tm 1   W  tm  2   
 um  2  um 1  um  W  tm  2   W  tm 3   
  u1  u2 
 um  W  t1 
3.3.2 Distribution of Brownian
Motion
  u1 , u2 ,
, um 
=  exp umW  tm   um 1W  tm 1   um  2W  t m  2  


 u1  t1 


  exp um W  t m   W  t m 1     exp  u m 1  u m  W  t m 1   W  t m  2   
 exp  u1  u2 
 um  W  t1 
2
1

1

 exp  um2  tm  tm 1    exp   um 1  um   tm 1  tm  2  
2

2

2 
1
exp   u1  u2   um  t1 
2

3.3.2 Distribution of Brownian
Motion
• The covariance matrix for Brownian motion
( i.e., for the m-dimensional random vector
W  t1  ,W  t2  , ,W  tm   ) is
  W 2  t1  
 W  t1 W  t 2  



2
  W  t2  W  t1  


W
 t2  






  W  tm  W  t1    W  t m W  t 2  
 W  t1 W  t m    t t
 1 1
 W  t 2 W  t m    t1 t 2

 


 W 2  t m    t1 t2
t1 
t2 



tm 
3.3.2 Distribution of Brownian
Motion
• Theorem 3.3.2 (Alternative characterizations
of Brownian motion)
Let  , F ,   be a probability space. For each
w   , suppose there is a continuous function
W  t  of t  0 that satisfies W  0   0 and that
depends on w . The following three properties
are equivalent.
3.3.2 Distribution of Brownian
Motion
1.
For all 0  t0  t1  …  tm , the increments
W  t1   W  t1   W  t0  , W  t 2   W  t1  ,
,W  t m   W  t m 1 
are independent and each of these
increments is normally distributed with
mean and variance given by
 W  ti 1   W  ti    0and
Var W  ti 1   W  ti    ti 1  ti .
3.3.2 Distribution of Brownian
Motion
2.
3.
For all 0  t0  t1  …  tm , the random
variables W  t1  ,W  t2  , ,W  tm  are jointly
normally distributed with means equal
to zero and covariance matrix.
For all 0  t0  t1  …  tm , the random
variables W  t1  ,W  t2  , ,W  tm  have the
joint moment-generating function.
3.3.2 Distribution of Brownian
Motion
• If any of 1, 2, or 3 holds ( and hence they all
hold), then is a Brownian motion.
3.3.3 Filtration for Brownian Motion
• Definition 3.3.3
Let  , F ,   be a probability space on which is
defined a Brownian motion W  t  , t  0.
A filtration for the Brownian motion is a
collection of  -algebra F  t  , t  0 satisfying：
3.3.3 Filtration for Brownian Motion
1.
( Information accumulates ) For 0  s  t ,
every set in F  s  is also in F  t  . In other
words, there is at least as much
information available at the later timeas
F  t  there is at the earlier time F  s  .
3.3.3 Filtration for Brownian Motion
2.
( Adaptivity ) For each t  0, the Brownian
motion W  t  at time t is F  t -measurable.
In other words, the information available
at time t is sufficient to evaluate the
Brownian motion W  t  at that time.
3.3.3 Filtration for Brownian Motion
3.
( Independence of future increments )
For 0  t  u, the increment W  u   W  t  is
independent of F  t . In other words, any
increment of the Brownian motion after
time t is independent of the information
available at time t.
3.3.3 Filtration for Brownian Motion
• Let  t  , t  0, be a stochastic process. We
say that  t  is adapted to the filtration F  t 
if for each t  0 the random variable  t  is
F  t  -measurable.
•
There are two possibilities for the
filtration F(t) for a Brownian motion：
1. F(t) contain only the information obtained
by observing the Brownian motion itself
up to time t.
2. F(t) information obtained by observing
the Browning motion and one or more
other process.
• Theorem 3.3.4
Brownian motion is a martingale.