Brownian Motion
Chuan-Hsiang Han
November 24, 2010
Symmetric Random Walk
Given Ω∞ , ℱ, 𝐏 ; let 𝜔 = 𝜔1 , 𝜔2 , 𝜔3 ⋯ ∈ Ω∞
𝟏
and 𝐏 𝐻 = 𝐏 𝑻 = , and 𝜔𝑛 denotes the
𝟐
outcome of 𝑛th toss. Define the r.v.'s
∞
𝑋𝑗
that for each 𝑗
𝑗=1
+1,
𝑋𝑗 =
−1,
𝑖𝑓 𝜔𝑗 = 𝐻
𝑖𝑓𝜔𝑗 = 𝑇
A S.R.W. is a process 𝑀𝑘 ∞
𝑘=0 such that 𝑀0 = 0
𝑘
and 𝑀𝑘 = 𝑗=1 𝑋𝑗 , 𝑘 = 1,2, ⋯ .
Independent Increments of S.R.W.
Choose 0 = 𝑘0 < 𝑘1 < ⋯ < 𝑘𝑚 , the r.v.s 𝑀𝑘1 =
𝑀𝑘1 − 𝑀𝑘0 , 𝑀𝑘2 − 𝑀𝑘1 , ⋯ 𝑀𝑘𝑚 − 𝑀𝑘𝑚−1 are
independent, where the increment is defined by
𝑘𝑖+1
𝑀𝑘𝑖+1 − 𝑀𝑘𝑖 = 𝑗=𝑘𝑖 +1 𝑋𝑗 .
Note:
(1) Increments are independent.
(2) The increment 𝑀𝑘𝑖+1 − 𝑀𝑘𝑖 has mean 0 and
variance 𝑘𝑖+1 − 𝑘𝑖 .(Stationarity)
Martingale Property of S.R.W.
For any nonnegative integers 𝑘 > 𝑙,
𝐸 𝑀𝑙 ℱ𝑘 = 𝐸 𝑀𝑙 − 𝑀𝑘 + 𝑀𝑘 ℱ𝑘 = 𝑀𝑘
ℱ𝑘 contains all the information of the first 𝑘 coin
tosses.
If R.W. is not symmetric, it is not a martingale.
Markov Property of S.R.W.
For any nonnegative integers 𝑘 > 𝑙 and any
integrable function 𝑓
𝐸 𝑓 𝑀𝑙 ℱ𝑘 = 𝐸 𝑓 𝑀𝑙 − 𝑀𝑘 + 𝑀𝑘 ℱ𝑘
𝑙−𝑘
=
𝐶 𝑙 − 𝑘, 𝑖
𝑖=0
1
2
𝑖
1
2
𝑙−𝑘−𝑖
𝑓 2𝑖 − 𝑙 + 𝑘
Quadratic Variation of S.R.W.
The quadratic variation up to time 𝑘 is defined
to be
𝑘
𝑀, 𝑀
𝑘
=
𝑀𝑗 − 𝑀𝑗−1
2
=𝑘
𝑗=1
Note the difference between 𝑉𝑎𝑟 𝑀𝑘 = 𝑘 (an
average over all paths), and 𝑀, 𝑀 𝑘 = 𝑘.
(pathwise property)
Scaled S.R.W.
Goal: to approximate Brownian Motion
1
𝑛
𝑊
𝑡𝑖 =
𝑀𝑛𝑡𝑖 𝑛𝑡𝑖 ∈𝑍+
𝑛
1
𝑛
1. new time interval is "very small" of instead of 1
2. magnitude is "small" of
1
𝑛
instead of 1.
For any 𝑡 ∈ 0, ∞ , 𝑊 𝑛 𝑡 can be defined as a
linear interpolation between the nearest 𝑡𝑖 such
that 𝑡𝑖 ≤ 𝑡 < 𝑡𝑖+1 .
Properties of Scaled S.R.W.
(i) independent increments: for any 0 = 𝑡0 < 𝑡1 < ⋯ <
𝑡𝑚 , 𝑊
𝑊
𝑛
𝑛
𝑡1 − 𝑊
𝑡2 − 𝑊
𝑛
𝑛
𝑡1
𝑡0
,
,⋯ 𝑊
𝑛
𝑡𝑚 −
(iii) Martingale property
𝐸 𝑊 𝑛 𝑡 ℱ𝑠 = 𝑊 𝑛 𝑠 .
(iv) Markov Property: for any function 𝑓, these exists a function 𝑔 so
that
𝐸 𝑓 𝑊 𝑛 𝑡 ℱ𝑠 = 𝑔 𝑊 𝑛 𝑠 .
(v) Quadratic variation: for any 𝑡 ≥ 0,
𝑛𝑡
𝑊
𝑛
𝑛𝑡
=
𝑗=1
,𝑊
𝑛
𝑡 =
𝑊
𝑗=1
1
= 𝑡.
𝑛
𝑛
𝑗
−𝑊
𝑛
𝑛
𝑗−1
𝑛
𝑛𝑡
2
=
𝑗=1
1
𝑋𝑗
𝑛
2
Limiting (Marginal) Distribution of
S.R.W.
Theorem 3.2.1. (Central Limit Theorem)
For any fixed 𝑡 ≥ 0,
𝑊
𝑛
𝑡
𝑛↑∞
𝑋 ≜ 𝑊𝑡 ~𝒩 0, 𝑡 in dist.
or
𝑃 𝑊
𝑛
𝑡 ≤𝑥
𝑛↑∞
𝑥
−∞
Proof: shown in class.
1
2𝜋𝑡
𝑒
−𝑧 2 2𝑡
𝑑𝑧 .
A Numerical Example
𝑃 𝜔: 0 ≤ 𝑊 100 0.25 ≤ 0.2
= 𝑃 𝜔: 0 ≤ 𝑀25 ≤ 2 = 0.1555
0.2
2 −2𝑧 2
𝑃 𝜔: 0 ≤ 𝑊 0.25 ≤ 0.2 =
𝑒
𝑑𝑧
2𝜋
0
≈ 0.1554
Log-Normality as the Limit of the
Binomial Model
Theorem 3.2.2. (Central Limit Theorem)
For any fixed 𝑡 ≥ 0,
𝑆𝑛 𝑡 = 𝑆 0
=𝑆 0
𝐻𝑛𝑡 𝑇𝑛𝑡 𝑛↑∞
𝑢𝑛 𝑑𝑛
𝑆
𝑡
𝜎2𝑡
−
−𝜎𝑊𝑡
2
𝑒
in the distribution sense, where 𝑢𝑛 = 1 +
𝑑𝑛 = 1 −
𝜎
,and
𝑛
𝑷 𝜔=𝐻 =
1+𝑟−𝑑𝑛
𝑢𝑛 −𝑑𝑛
𝜎
,
𝑛
What is Brownian Motion?
"If 𝑊 is a continuous process with independent
increments that are normally distributed, then
𝑊 is a Brownian motion."
Standard Brownian Motions
check Definition 3.3.1 in the text.
Definition of SBM: Let the stochastic process
𝑊𝑡 , 𝑡 ≥ 0 under a probability space
Ω, ℱ, P be continuous and satisfy:
1. 𝑊0 = 0
2. 𝑊𝑡+𝑠 − 𝑊𝑡 ~𝒩 0, 𝑠
3. 𝑊𝑡+𝑠 − 𝑊𝑡 is independent of 𝑊𝑡𝑖 − 𝑊𝑡𝑖+1 for
𝑡0 < ⋯ 𝑡𝑛 = 𝑡.
Covariance Matrix
Check 𝐶𝑜𝑣 𝑊𝑠 , 𝑊𝑡 = 𝑚𝑖𝑛 𝑠, 𝑡 for any
nonnegative 𝑠 and 𝑡
𝑇
For any vector 𝑉 = 𝑊𝑡1 , 𝑊𝑡2 , ⋯ , 𝑊𝑡𝑚 with
0 ≤ 𝑡1 ≤ ⋯ ≤ 𝑡𝑚 ,
𝑡1 𝑡1 ⋯ 𝑡1
𝑡1 𝑡2 ⋯ 𝑡2
𝑇
𝐶𝑜𝑣 𝑉𝑉 =
≜𝐶
⋮
⋮ ⋮
𝑡1 𝑡2 ⋯ 𝑡𝑚
In fact, 𝑉~𝒩 0, 𝐶 .
Joint Moment-Generating Function of BM
𝜑 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑚 = 𝐸 𝑒𝑥𝑝 𝑢 ∙ 𝑉
= 𝐸 𝑒𝑥𝑝 𝑢1 𝑊𝑡1 + 𝑢2 𝑊𝑡2 + ⋯ + 𝑢𝑚 𝑊𝑡𝑚
1
= 𝑒𝑥𝑝
𝑢1 + 𝑢2 + ⋯ + 𝑢𝑚 2 𝑡1
2
1
+ 𝑢2 + ⋯ + 𝑢𝑚 2 𝑡2 − 𝑡1 + ⋯
2
Alternative Characteristics of Brownian
Motion (Theorem 3.3.2)
For any continuous process 𝑊𝑡 , 𝑡 ≥ 0 with 𝑊0 = 0,
the following three properties are equivalent.
(i) increments are independent and normally
distributed.
(ii) For any 0 ≤ 𝑡0 ≤ 𝑡1 ≤ ⋯ ≤ 𝑡𝑚 ,
𝑊𝑡1 , 𝑊𝑡2 , ⋯ , 𝑊𝑡𝑚 are jointly normally distributed.
(ii) 𝑊𝑡1 , 𝑊𝑡2 , ⋯ , 𝑊𝑡𝑚 has the joint momentgenerating function as before.
If any of the three holds, then 𝑊𝑡 , 𝑡 ≥ 0, is a SBM.
Filtration for B.M.
Definition 3.3.3 Let Ω, ℱ, 𝑷 be a probability space
on which the B.M. 𝑊𝑡 𝑡≥0 is defined. A filtration
for the B.M. is a collection of 𝜎-algebras ℱ𝑡 𝑡≥0 ,
satisfying
(i) (Information accumulates) For 0 ≤ 𝑠 ≤ 𝑡, ℱ𝑠 ⊆
ℱ𝑡 .
(ii) (Adaptivity) each 𝑊𝑡 is ℱ𝑡 -measurable.
(iii) (Independence of future increments) 0 ≤ 𝑡 < 𝑢 ,
the increment 𝑊𝑢 − 𝑊𝑡 is independent of ℱ𝑡 . [Note,
this property leads to Efficient Market Hypothesis.]
Martingale property
Theorem 3.3.4 B.M. is a martingale.
Proof:
𝐸 𝑊𝑡 |ℱ𝑠 = ⋯ = 𝑊𝑠
Levy's Characteristics of Brownian
Motion
The process 𝑊𝑡 is SBM iff the conditional
characterization function is
𝐸 𝑒 𝑖𝑢
𝑊𝑡 −𝑊𝑥
|ℱ𝑠 =
𝑢2 𝑡−𝑠
−
2
𝑒
Variations: First-Order (Total) Variation
Given a function 𝑓 defined on 0, 𝑇 , the total
variation 𝑇𝑉𝑇 𝑓 is defined by
𝑛−1
𝑇𝑉𝑇 𝑓 = lim
Π →0
𝑓 𝑡𝑗+1 − 𝑓 𝑡𝑗
𝑗=0
where the partition Π = 𝑡0 = 0, 𝑡1 , ⋯ , 𝑡𝑛 = 𝑇
and Π = 𝑚𝑎𝑥𝑖=0,⋯,𝑛−1 𝑡𝑗+1 − 𝑡𝑗
If 𝑓 is differentiable,
𝑓 𝑡𝑗+1 − 𝑓 𝑡𝑗 = 𝑓′ 𝑡𝑗⋆ 𝑡𝑗+1 − 𝑡𝑗
for some 𝑡𝑗⋆ 𝜖 𝑡𝑗+1 , 𝑡𝑗 . Then
𝑇𝑉𝑇 𝑓 = lim
lim
Π →0
Π →0
𝑛−1
⋆
𝑓′
𝑡
𝑗
𝑗=0
𝑛−1
𝑗=0
𝑓 𝑡𝑗+1 − 𝑓 𝑡𝑗
𝑡𝑗+1 − 𝑡𝑗 =
𝑇
0
=
𝑓′ 𝑡 𝑑𝑡.
Quadratic Variation
Def. 3.4.1 The quadratic variation of 𝑓 up to
time 𝑇 is defined by
𝑛−1
𝑓, 𝑓
𝑇
= lim
Π →0
𝑓 𝑡𝑗+1 − 𝑓 𝑡𝑗
𝑗=0
2
If 𝑓 is continuous differentiable,
𝑓 𝑡𝑗+1 − 𝑓 𝑡𝑗 = 𝑓′ 𝑡𝑗⋆ 𝑡𝑗+1 − 𝑡𝑗
for some 𝑡𝑗⋆ 𝜖 𝑡𝑗+1 , 𝑡𝑗 . Then
𝑓, 𝑓 𝑇 = lim 𝑛−1
𝑓 𝑡𝑗+1
𝑗=0
Π →0
𝑛−1
⋆ 2
lim 𝑗=0 𝑓′ 𝑡𝑗
𝑡𝑗+1 − 𝑡𝑗
Π →0
𝑇
2
𝑓′
𝑡
𝑑𝑡
0
− 𝑓 𝑡𝑗
= lim
2
Π →0
≤
Π ×
Quadratic Variation of B.M.
Thm. 3.4.3 Let 𝑊𝑡≥0 be a Brownian Motion.
Then 𝑊, 𝑊 𝑇 = 𝑇 for all 𝑇 ≥ 0 a.s..
B.M. accumulates quadratic variation at rate one
per unit time.
Informal notion:
𝑑𝑊𝑡 ∙ 𝑑𝑊𝑡 = 𝑑𝑡, 𝑑𝑊𝑡 ∙ 𝑑𝑡 = 0, 𝑑𝑡 ∙ 𝑑𝑡 = 0
Geometric Brownian Motion
The geometric Brownian motion is a process of the
following form
𝑆𝑡 = 𝑆0 𝑒𝑥𝑝 𝜎𝑊𝑡 + 𝜇 − 𝜎 2 2 𝑡 .
where 𝑆0 is the current value, 𝑊𝑡≥0 is a B.M., 𝜇 is the drift
and 𝜎 > 0 is the volatility.
For each partition 𝑡0 = 0, 𝑡1 , ⋯ , 𝑡𝑛 = 𝑇 , define the log
returns
𝑆𝑡𝑗+1
𝑙𝑜𝑔
= 𝜎 𝑊𝑡𝑗+1 − 𝑊𝑡𝑗 + 𝜇 − 𝜎 2 2 𝑡𝑗+1 − 𝑡𝑗
𝑆𝑡𝑗
Volatility Estimation of GBM
The realized variance is defined by
2
𝑛−1
𝑆𝑡𝑗+1
𝑙𝑜𝑔
𝑆𝑡𝑗
𝑗=0
which converges to 𝜎 2 𝑇 as Π → 0
BM is a Markov process
Thm. 3.5.1 Let 𝑊𝑡≥0 be a B.M. and ℱ𝑡≥0 be a
filtration for this B.M.. Then
(1)Wt0 is a Markov process.
Thm. 3.6.1.
(2)𝑍𝑡 = 𝑒𝑥𝑝 𝜎𝑊𝑡 −
1 2
𝜎 𝑡
2
is martingale.
(We call 𝑍𝑡 exponential martingale.)
Note: 𝐸 𝑍𝑡 = 1