Brownian Motion
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Transcript Brownian Motion
Brownian Motion
Chuan-Hsiang Han
November 24, 2010
Symmetric Random Walk
Given ฮฉโ , โฑ, ๐ ; let ๐ = ๐1 , ๐2 , ๐3 โฏ โ ฮฉโ
๐
and ๐ ๐ป = ๐ ๐ป = , and ๐๐ denotes the
๐
outcome of ๐th toss. Define the r.v.'s
โ
๐๐
that for each ๐
๐=1
+1,
๐๐ =
โ1,
๐๐ ๐๐ = ๐ป
๐๐๐๐ = ๐
A S.R.W. is a process ๐๐ โ
๐=0 such that ๐0 = 0
๐
and ๐๐ = ๐=1 ๐๐ , ๐ = 1,2, โฏ .
Independent Increments of S.R.W.
Choose 0 = ๐0 < ๐1 < โฏ < ๐๐ , the r.v.s ๐๐1 =
๐๐1 โ ๐๐0 , ๐๐2 โ ๐๐1 , โฏ ๐๐๐ โ ๐๐๐โ1 are
independent, where the increment is defined by
๐๐+1
๐๐๐+1 โ ๐๐๐ = ๐=๐๐ +1 ๐๐ .
Note:
(1) Increments are independent.
(2) The increment ๐๐๐+1 โ ๐๐๐ has mean 0 and
variance ๐๐+1 โ ๐๐ .(Stationarity)
Martingale Property of S.R.W.
For any nonnegative integers ๐ > ๐,
๐ธ ๐๐ โฑ๐ = ๐ธ ๐๐ โ ๐๐ + ๐๐ โฑ๐ = ๐๐
โฑ๐ contains all the information of the first ๐ coin
tosses.
If R.W. is not symmetric, it is not a martingale.
Markov Property of S.R.W.
For any nonnegative integers ๐ > ๐ and any
integrable function ๐
๐ธ ๐ ๐๐ โฑ๐ = ๐ธ ๐ ๐๐ โ ๐๐ + ๐๐ โฑ๐
๐โ๐
=
๐ถ ๐ โ ๐, ๐
๐=0
1
2
๐
1
2
๐โ๐โ๐
๐ 2๐ โ ๐ + ๐
Quadratic Variation of S.R.W.
The quadratic variation up to time ๐ is defined
to be
๐
๐, ๐
๐
=
๐๐ โ ๐๐โ1
2
=๐
๐=1
Note the difference between ๐๐๐ ๐๐ = ๐ (an
average over all paths), and ๐, ๐ ๐ = ๐.
(pathwise property)
Scaled S.R.W.
Goal: to approximate Brownian Motion
1
๐
๐
๐ก๐ =
๐๐๐ก๐ ๐๐ก๐ โ๐+
๐
1
๐
1. new time interval is "very small" of instead of 1
2. magnitude is "small" of
1
๐
instead of 1.
For any ๐ก โ 0, โ , ๐ ๐ ๐ก can be defined as a
linear interpolation between the nearest ๐ก๐ such
that ๐ก๐ โค ๐ก < ๐ก๐+1 .
Properties of Scaled S.R.W.
(i) independent increments: for any 0 = ๐ก0 < ๐ก1 < โฏ <
๐ก๐ , ๐
๐
๐
๐
๐ก1 โ ๐
๐ก2 โ ๐
๐
๐
๐ก1
๐ก0
,
,โฏ ๐
๐
๐ก๐ โ
(iii) Martingale property
๐ธ ๐ ๐ ๐ก โฑ๐ = ๐ ๐ ๐ .
(iv) Markov Property: for any function ๐, these exists a function ๐ so
that
๐ธ ๐ ๐ ๐ ๐ก โฑ๐ = ๐ ๐ ๐ ๐ .
(v) Quadratic variation: for any ๐ก โฅ 0,
๐๐ก
๐
๐
๐๐ก
=
๐=1
,๐
๐
๐ก =
๐
๐=1
1
= ๐ก.
๐
๐
๐
โ๐
๐
๐
๐โ1
๐
๐๐ก
2
=
๐=1
1
๐๐
๐
2
Limiting (Marginal) Distribution of
S.R.W.
Theorem 3.2.1. (Central Limit Theorem)
For any fixed ๐ก โฅ 0,
๐
๐
๐ก
๐โโ
๐ โ ๐๐ก ~๐ฉ 0, ๐ก in dist.
or
๐ ๐
๐
๐ก โค๐ฅ
๐โโ
๐ฅ
โโ
Proof: shown in class.
1
2๐๐ก
๐
โ๐ง 2 2๐ก
๐๐ง .
A Numerical Example
๐ ๐: 0 โค ๐ 100 0.25 โค 0.2
= ๐ ๐: 0 โค ๐25 โค 2 = 0.1555
0.2
2 โ2๐ง 2
๐ ๐: 0 โค ๐ 0.25 โค 0.2 =
๐
๐๐ง
2๐
0
โ 0.1554
Log-Normality as the Limit of the
Binomial Model
Theorem 3.2.2. (Central Limit Theorem)
For any fixed ๐ก โฅ 0,
๐๐ ๐ก = ๐ 0
=๐ 0
๐ป๐๐ก ๐๐๐ก ๐โโ
๐ข๐ ๐๐
๐
๐2 ๐ก
โ
โ๐๐๐ก
2
๐
๐ก
in the distribution sense, where ๐ข๐ = 1 +
๐๐ = 1 โ
๐
,and
๐
๐ท ๐=๐ป =
1+๐โ๐๐
๐ข๐ โ๐๐
๐
,
๐
What is Brownian Motion?
"If ๐ is a continuous process with independent
increments that are normally distributed, then
๐ is a Brownian motion."
Standard Brownian Motions
check Definition 3.3.1 in the text.
Definition of SBM: Let the stochastic process
๐๐ก , ๐ก โฅ 0 under a probability space
ฮฉ, โฑ, P be continuous and satisfy:
1. ๐0 = 0
2. ๐๐ก+๐ โ ๐๐ก ~๐ฉ 0, ๐
3. ๐๐ก+๐ โ ๐๐ก is independent of ๐๐ก๐ โ ๐๐ก๐+1 for
๐ก0 < โฏ ๐ก๐ = ๐ก.
Covariance Matrix
Check ๐ถ๐๐ฃ ๐๐ , ๐๐ก = ๐๐๐ ๐ , ๐ก for any
nonnegative ๐ and ๐ก
๐
For any vector ๐ = ๐๐ก1 , ๐๐ก2 , โฏ , ๐๐ก๐ with
0 โค ๐ก1 โค โฏ โค ๐ก๐ ,
๐ก1 ๐ก1 โฏ ๐ก1
๐ก1 ๐ก2 โฏ ๐ก2
๐
๐ถ๐๐ฃ ๐๐ =
โ๐ถ
โฎ
โฎ โฎ
๐ก1 ๐ก2 โฏ ๐ก๐
In fact, ๐~๐ฉ 0, ๐ถ .
Joint Moment-Generating Function of BM
๐ ๐ข1 , ๐ข2 , โฏ , ๐ข๐ = ๐ธ ๐๐ฅ๐ ๐ข โ ๐
= ๐ธ ๐๐ฅ๐ ๐ข1 ๐๐ก1 + ๐ข2 ๐๐ก2 + โฏ + ๐ข๐ ๐๐ก๐
1
= ๐๐ฅ๐
๐ข1 + ๐ข2 + โฏ + ๐ข๐ 2 ๐ก1
2
1
+ ๐ข2 + โฏ + ๐ข๐ 2 ๐ก2 โ ๐ก1 + โฏ
2
Alternative Characteristics of Brownian
Motion (Theorem 3.3.2)
For any continuous process ๐๐ก , ๐ก โฅ 0 with ๐0 = 0,
the following three properties are equivalent.
(i) increments are independent and normally
distributed.
(ii) For any 0 โค ๐ก0 โค ๐ก1 โค โฏ โค ๐ก๐ ,
๐๐ก1 , ๐๐ก2 , โฏ , ๐๐ก๐ are jointly normally distributed.
(ii) ๐๐ก1 , ๐๐ก2 , โฏ , ๐๐ก๐ has the joint momentgenerating function as before.
If any of the three holds, then ๐๐ก , ๐ก โฅ 0, is a SBM.
Filtration for B.M.
Definition 3.3.3 Let ฮฉ, โฑ, ๐ท be a probability space
on which the B.M. ๐๐ก ๐กโฅ0 is defined. A filtration
for the B.M. is a collection of ๐-algebras โฑ๐ก ๐กโฅ0 ,
satisfying
(i) (Information accumulates) For 0 โค ๐ โค ๐ก, โฑ๐ โ
โฑ๐ก .
(ii) (Adaptivity) each ๐๐ก is โฑ๐ก -measurable.
(iii) (Independence of future increments) 0 โค ๐ก < ๐ข ,
the increment ๐๐ข โ ๐๐ก is independent of โฑ๐ก . [Note,
this property leads to Efficient Market Hypothesis.]
Martingale property
Theorem 3.3.4 B.M. is a martingale.
Proof:
๐ธ ๐๐ก |โฑ๐ = โฏ = ๐๐
Levy's Characteristics of Brownian
Motion
The process ๐๐ก is SBM iff the conditional
characterization function is
๐ธ ๐ ๐๐ข
๐๐ก โ๐๐ฅ
|โฑ๐ =
๐ข2 ๐กโ๐
โ
2
๐
Variations: First-Order (Total) Variation
Given a function ๐ defined on 0, ๐ , the total
variation ๐๐๐ ๐ is defined by
๐โ1
๐๐๐ ๐ = lim
ฮ โ0
๐ ๐ก๐+1 โ ๐ ๐ก๐
๐=0
where the partition ฮ = ๐ก0 = 0, ๐ก1 , โฏ , ๐ก๐ = ๐
and ฮ = ๐๐๐ฅ๐=0,โฏ,๐โ1 ๐ก๐+1 โ ๐ก๐
If ๐ is differentiable,
๐ ๐ก๐+1 โ ๐ ๐ก๐ = ๐โฒ ๐ก๐โ ๐ก๐+1 โ ๐ก๐
for some ๐ก๐โ ๐ ๐ก๐+1 , ๐ก๐ . Then
๐๐๐ ๐ = lim
lim
ฮ โ0
ฮ โ0
๐โ1
โ
๐โฒ
๐ก
๐
๐=0
๐โ1
๐=0
๐ ๐ก๐+1 โ ๐ ๐ก๐
๐ก๐+1 โ ๐ก๐ =
๐
0
=
๐โฒ ๐ก ๐๐ก.
Quadratic Variation
Def. 3.4.1 The quadratic variation of ๐ up to
time ๐ is defined by
๐โ1
๐, ๐
๐
= lim
ฮ โ0
๐ ๐ก๐+1 โ ๐ ๐ก๐
๐=0
2
If ๐ is continuous differentiable,
๐ ๐ก๐+1 โ ๐ ๐ก๐ = ๐โฒ ๐ก๐โ ๐ก๐+1 โ ๐ก๐
for some ๐ก๐โ ๐ ๐ก๐+1 , ๐ก๐ . Then
๐, ๐ ๐ = lim ๐โ1
๐ ๐ก๐+1
๐=0
ฮ โ0
๐โ1
โ 2
lim ๐=0 ๐โฒ ๐ก๐
๐ก๐+1 โ ๐ก๐
ฮ โ0
๐
2
๐โฒ
๐ก
๐๐ก
0
โ ๐ ๐ก๐
= lim
2
ฮ โ0
โค
ฮ ×
Quadratic Variation of B.M.
Thm. 3.4.3 Let ๐๐กโฅ0 be a Brownian Motion.
Then ๐, ๐ ๐ = ๐ for all ๐ โฅ 0 a.s..
B.M. accumulates quadratic variation at rate one
per unit time.
Informal notion:
๐๐๐ก โ ๐๐๐ก = ๐๐ก, ๐๐๐ก โ ๐๐ก = 0, ๐๐ก โ ๐๐ก = 0
Geometric Brownian Motion
The geometric Brownian motion is a process of the
following form
๐๐ก = ๐0 ๐๐ฅ๐ ๐๐๐ก + ๐ โ ๐ 2 2 ๐ก .
where ๐0 is the current value, ๐๐กโฅ0 is a B.M., ๐ is the drift
and ๐ > 0 is the volatility.
For each partition ๐ก0 = 0, ๐ก1 , โฏ , ๐ก๐ = ๐ , define the log
returns
๐๐ก๐+1
๐๐๐
= ๐ ๐๐ก๐+1 โ ๐๐ก๐ + ๐ โ ๐ 2 2 ๐ก๐+1 โ ๐ก๐
๐๐ก๐
Volatility Estimation of GBM
The realized variance is defined by
2
๐โ1
๐๐ก๐+1
๐๐๐
๐๐ก๐
๐=0
which converges to ๐ 2 ๐ as ฮ โ 0
BM is a Markov process
Thm. 3.5.1 Let ๐๐กโฅ0 be a B.M. and โฑ๐กโฅ0 be a
filtration for this B.M.. Then
(1)Wt0 is a Markov process.
Thm. 3.6.1.
(2)๐๐ก = ๐๐ฅ๐ ๐๐๐ก โ
1 2
๐ ๐ก
2
is martingale.
(We call ๐๐ก exponential martingale.)
Note: ๐ธ ๐๐ก = 1