### Brownian Motion

```Brownian Motion
Chuan-Hsiang Han
November 24, 2010
Symmetric Random Walk
Given Ω∞ , ℱ,  ; let  = 1 , 2 , 3 ⋯ ∈ Ω∞

and   =   = , and  denotes the

outcome of th toss. Define the r.v.'s
∞

that for each
=1
+1,
=
−1,
=
=
A S.R.W. is a process  ∞
=0 such that 0 = 0

and  = =1  ,  = 1,2, ⋯ .
Independent Increments of S.R.W.
Choose 0 = 0 < 1 < ⋯ <  , the r.v.s 1 =
1 − 0 , 2 − 1 , ⋯  − −1 are
independent, where the increment is defined by
+1
+1 −  = = +1  .
Note:
(1) Increments are independent.
(2) The increment +1 −  has mean 0 and
variance +1 −  .(Stationarity)
Martingale Property of S.R.W.
For any nonnegative integers  > ,
ℱ =   −  +  ℱ =
ℱ contains all the information of the first  coin
tosses.
If R.W. is not symmetric, it is not a martingale.
Markov Property of S.R.W.
For any nonnegative integers  >  and any
integrable function
ℱ =    −  +  ℱ
−
=
− ,
=0
1
2

1
2
−−
2 −  +
The quadratic variation up to time  is defined
to be

,

=
− −1
2
=
=1
Note the difference between   =  (an
average over all paths), and ,   = .
(pathwise property)
Scaled S.R.W.
Goal: to approximate Brownian Motion
1

=
∈+

1

1. new time interval is "very small" of instead of 1
2. magnitude is "small" of
1

For any  ∈ 0, ∞ ,    can be defined as a
linear interpolation between the nearest  such
that  ≤  < +1 .
Properties of Scaled S.R.W.
(i) independent increments: for any 0 = 0 < 1 < ⋯ <
,

1 −
2 −

1
0
,
,⋯

−
(iii) Martingale property
ℱ =    .
(iv) Markov Property: for any function , these exists a function  so
that
ℱ =     .
(v) Quadratic variation: for any  ≥ 0,

=
=1
,

=

=1
1
= .

−

−1

2
=
=1
1

2
Limiting (Marginal) Distribution of
S.R.W.
Theorem 3.2.1. (Central Limit Theorem)
For any fixed  ≥ 0,

↑∞
≜  ~ 0,  in dist.
or

≤
↑∞

−∞
Proof: shown in class.
1
2

− 2 2
.
A Numerical Example
: 0 ≤  100 0.25 ≤ 0.2
=  : 0 ≤ 25 ≤ 2 = 0.1555
0.2
2 −2 2
: 0 ≤  0.25 ≤ 0.2 =

2
0
≈ 0.1554
Log-Normality as the Limit of the
Binomial Model
Theorem 3.2.2. (Central Limit Theorem)
For any fixed  ≥ 0,
=  0
= 0
↑∞

2
−
−
2

in the distribution sense, where  = 1 +
= 1 −

,and

= =
1+−
−

,

What is Brownian Motion?
"If  is a continuous process with independent
increments that are normally distributed, then
is a Brownian motion."
Standard Brownian Motions
check Definition 3.3.1 in the text.
Definition of SBM: Let the stochastic process
,  ≥ 0 under a probability space
Ω, ℱ, P be continuous and satisfy:
1. 0 = 0
2. + −  ~ 0,
3. + −  is independent of  − +1 for
0 < ⋯  = .
Covariance Matrix
Check   ,  =  ,  for any
nonnegative  and

For any vector  = 1 , 2 , ⋯ ,  with
0 ≤ 1 ≤ ⋯ ≤  ,
1 1 ⋯ 1
1 2 ⋯ 2

=
≜
⋮
⋮ ⋮
1 2 ⋯
In fact, ~ 0,  .
Joint Moment-Generating Function of BM
1 , 2 , ⋯ ,  =    ∙
=   1 1 + 2 2 + ⋯ +
1
=
1 + 2 + ⋯ +  2 1
2
1
+ 2 + ⋯ +  2 2 − 1 + ⋯
2
Alternative Characteristics of Brownian
Motion (Theorem 3.3.2)
For any continuous process  ,  ≥ 0 with 0 = 0,
the following three properties are equivalent.
(i) increments are independent and normally
distributed.
(ii) For any 0 ≤ 0 ≤ 1 ≤ ⋯ ≤  ,
1 , 2 , ⋯ ,  are jointly normally distributed.
(ii) 1 , 2 , ⋯ ,  has the joint momentgenerating function as before.
If any of the three holds, then  ,  ≥ 0, is a SBM.
Filtration for B.M.
Definition 3.3.3 Let Ω, ℱ,  be a probability space
on which the B.M.  ≥0 is defined. A filtration
for the B.M. is a collection of -algebras ℱ ≥0 ,
satisfying
(i) (Information accumulates) For 0 ≤  ≤ , ℱ ⊆
ℱ .
(ii) (Adaptivity) each  is ℱ -measurable.
(iii) (Independence of future increments) 0 ≤  <  ,
the increment  −  is independent of ℱ . [Note,
this property leads to Efficient Market Hypothesis.]
Martingale property
Theorem 3.3.4 B.M. is a martingale.
Proof:
|ℱ = ⋯ =
Levy's Characteristics of Brownian
Motion
The process  is SBM iff the conditional
characterization function is

−
|ℱ =
2 −
−
2

Variations: First-Order (Total) Variation
Given a function  defined on 0,  , the total
variation   is defined by
−1
= lim
Π →0
+1 −
=0
where the partition Π = 0 = 0, 1 , ⋯ ,  =
and Π = =0,⋯,−1 +1 −
If  is differentiable,
+1 −   = ′ ⋆ +1 −
for some ⋆  +1 ,  . Then
= lim
lim
Π →0
Π →0
−1
⋆
′

=0
−1
=0
+1 −
+1 −  =

0
=
′  .
Def. 3.4.1 The quadratic variation of  up to
time  is defined by
−1
,

= lim
Π →0
+1 −
=0
2
If  is continuous differentiable,
+1 −   = ′ ⋆ +1 −
for some ⋆  +1 ,  . Then
,   = lim −1
+1
=0
Π →0
−1
⋆ 2
lim =0 ′
+1 −
Π →0

2
′

0
−
= lim
2
Π →0
≤
Π ×
Thm. 3.4.3 Let ≥0 be a Brownian Motion.
Then ,   =  for all  ≥ 0 a.s..
B.M. accumulates quadratic variation at rate one
per unit time.
Informal notion:
∙  = ,  ∙  = 0,  ∙  = 0
Geometric Brownian Motion
The geometric Brownian motion is a process of the
following form
= 0   +  −  2 2  .
where 0 is the current value, ≥0 is a B.M.,  is the drift
and  > 0 is the volatility.
For each partition 0 = 0, 1 , ⋯ ,  =  , define the log
returns
+1

=  +1 −  +  −  2 2 +1 −

Volatility Estimation of GBM
The realized variance is defined by
2
−1
+1

=0
which converges to  2  as Π → 0
BM is a Markov process
Thm. 3.5.1 Let ≥0 be a B.M. and ℱ≥0 be a
filtration for this B.M.. Then
(1)Wt0 is a Markov process.
Thm. 3.6.1.
(2) =   −
1 2

2
is martingale.
(We call  exponential martingale.)
Note:   = 1
```