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ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Constructing Brownian Motions andRadon-Nikodym Derivatives
Steven R. Dunbar
February 5, 2016
1 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Outline
1 Standard Brownian Motion
2 Binomial Trees
3 Using the Radon-Nikodym derivative
2 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Comment
I tend to use Brownian Motion and Wiener Processinterchangeably, but
I like to use Brownian Motion for a physicalmanifestation (stock prices, motion of a particle in afluid)I like to use Wiener Process for the mathematicalmodel.
3 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Definition
Wiener ProcessThe Standard Wiener Process is a stochastic processW (t), for t ≥ 0, with the following properties:
1 Increments W (t)−W (s) are normally distributedW (t)−W (s) ∼ N(0, t− s).
2 For t1 < t2 ≤ t3 < t4, increments W (t4)−W (t3)and W (t2)−W (t1) are independent randomvariables.
3 W (0) = 0.4 W (t) is continuous for all t (with probability 1).
4 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Random Walk
Let
Yi =
+1 with probability 1/2−1 with probability 1/2
be a sequence of
independent, identically distributed Bernoulli randomvariables. Let Y0 = 0 for convenience and let
Tn =n∑
i=0
Yi
Note that Var [Yi] = 1, which we will need to use in amoment.
5 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Random Walk Graph
n 0 1 2 3 4 5 6 7 8 9 10Yn 0 1 1 1 -1 -1 -1 1 -1 -1 -1Tn 0 1 2 3 2 1 0 1 -1 -2 -3
0 2 4 6 8 10-2
-1
0
1
2
3
6 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Random Walk as continuous function
Sketch the random fortune Tn versus time using linearinterpolation between the points (n− 1, Tn−1) and(n, Tn).
The interpolation defines a function W (t) defined on[0,∞) with W (n) = Tn.
The notation W (t) reminds us of the piecewise linearnature of the fu
7 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Random Walk Continuous FunctionExample
0 2 4 6 8 10-2
-1
0
1
2
3WcaretN
8 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Scaling Random Walk
Compress time, rescale the space in a connected way. LetN be a large integer, and consider the rescaled function
WN(t) =
(1√N
)W (Nt).
This has the effect of taking a step of size ±1/√N in
1/N time unit. For example,
WN(1/N) =
(1√N
)W (N · 1/N) =
T1√N
=Y1√N.
9 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Distribution of the Scaled Walk
Now consider
WN(1) =W (N · 1)√
N=W (N)√
N=
TN√N.
According to the Central Limit Theorem, this quantity isapproximately normally distributed, with mean zero, andvariance 1. More generally,
WN(t) =W (Nt)√
N=√tW (Nt)√
Nt
If Nt is an integer, WN(t) is normally distributed withmean 0 and variance t. Furthermore, WN(0) = 0 andWN(t) is a continuous function.10 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Limit Theorem
Theorem (essentially Donsker, 1951)The limit of the rescaled random walk definingapproximate Brownian Motion is Brownian Motion in thefollowing sense:
P[WN(t1) < x1, WN(t2) < x2, . . . WN(tn) < xn
]→
P [W (t1) < x1,W (t2) < x2, . . .W (tn) < xn]
as N →∞ where t1 < t2 < · · · < tn.
11 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Example
Octave Source (compressed for space)p = 0.5;global N = 400; global T = 1; global SS = zeros(N+1, 1);S(2:N+1) = cumsum( 2 * (rand(N,1)<=p) - 1);function retval = WcaretN(x)
global N; global T; global S;Delta = T/N;prior = floor(x/Delta) + 1; # add 1 since arrays are 1basedsubsequent = ceil(x/Delta) + 1;retval = sqrt(Delta)*(S(prior) + ((x/Delta+1) - prior).*(S(subsequent)-S(prior)));
endfunctionfplot(@WcaretN, [0,T])
12 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Example
13 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
The goal
Build more intuition about Brownian Motions by lookingat probability on several discrete binomial trees.
The goal is to be very strongly intuitive and motivationalin contrast to rigorous, but still to do honest examples.
14 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
A standard binomial tree
Note that the expected value at time n is 0.
H
1/2
T
1/2
HH
1/2
HT
1/2
TH1/2
TT
1/2
HHH
1/2
HHT
1/2
HTH1/2
HTT
1/2
THH1/2
THT
1/2
TTH1/2
TTT
1/2
15 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
A skewed binomial tree
The expected value at n is 13n, this process is “drifting”
upward.
H
2/3
T
1/3
HH
2/3
HT
1/3
TH2/3
TT
1/3
HHH
2/3
HHT
1/3
HTH2/3
HTT
1/3
THH2/3
THT
1/3
TTH2/3
TTT
1/316 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
A general binomial tree
Instead of branch probabilities, give the path probabilitymeasure instead. Each can be recovered from the other.
πH
p11
πT
p10
πHH
p21
πHT
p20
πTHp21
πTT
p20
πHHH
p31
πHHT
p30
πHTHp31
πHTT
p30
πTHHp31
πTHT
p30
πTTHp31
πTTT
p3017 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Another general binomial tree
Another probability measure. What is the relation to theprevious measure?
0
φH
φT
φHH
φHT
φTH
φTT
φHHH
φHHT
φHTH
φHTT
φTHH
φTHT
φTTH
φTTT
18 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
The Radon-Nikodym derivative on the tree
The Radon-Nikodym derivative measures the likelihoodratio.
φH⁄πH
φT ⁄πT
φHH⁄πHH
φHT⁄πHT
φTH⁄πTH
φTT ⁄πTT
φHHH⁄πHHH
φHHT ⁄πHHT
φHTH ⁄πHTH
φHTT ⁄πHTT
φTHH ⁄πTHH
φTHT ⁄πTHT
φTTH⁄πTTH
φTTT ⁄πTTT
19 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Recover φ from π
Given P on paths, and the R-N derivative dQdP ,
Q =dQdP
P
Note also: the R-N derivative dQdP is defined on paths, so
it too is a random variable on the space Ω.
20 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Equivalent Measures
Equivalent measuresTwo probability measures on the space Ω with σ-algebraF are equivalent if for any set B ∈ F , P [B] > 0 if andonly if Q [B] > 0. This is, the probability measures areequivalent if P Q and Q P.
21 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Expectation
The Radon-Nikodym derivative is defined on paths, andis F -measurable, so it is also a random variable.
Even more, the Radon-Nikodym derivative is Ft-adapted.
Let ζt = dQdP folowing paths to time t.
ζt = EP
[dQdP|Ft
]for every t.
The expectation, knowing the information up to time tmeasured by P, represents the amount of change ofmeasure so far up to time t along the current path as ζt.
22 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Using the R-N Derivative
If we want to know EQ [F (Xt)], it would be EP [ζtF (Xt)].
If we want to know EQ [F (Xt)|Fs], then we would needthe amount of change from time s to t which is justζt/ζs, which is change up to time t with the change upto time s removed. In other words
EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .
23 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Passing to the Limit
Suppose P and Q are equivalent measures on a tree ofpaths. Given scaled path points (t1, x1), . . . (tn, xn) withtn = 1 then dQ
dP up to time T = 1 is the limit of thelikelihood ratios:
dQdP
= limn→∞
fnQ(x1, . . . , xn)
fnP (x1, . . . , xn)
Then for Brownian Motions let ζt = dQdP . Then
ζt = EP
[dQdP|Ft
]and
EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .24 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Simple Example
The point of the example is to apply the defintions andnotations to the pair of binomial trees with P defined byP [H] = 1
2and P [T ] = 1
2and Q [H] = 2
3and Q [T ] = 1
3.
This has minimal mathematical content, but is a goodillustration of defintions and notation.
ζt =
(4
3
) t+Xt2
·(
2
3
) t−Xt2
25 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Simple Example Continued
EQ[1[X3≥1]|F2
]Node Expectation X2 ProbHH 2
3· 1 + 1
3· 1 2 1
HT 23· 1 + 1
3· 0 1 2
3
TH 23· 1 + 1
3· 0 1
TT 23· 0 + 1
3· 0 0 0
Conditional probabilities the “old-fashioned way”
EQ[1[X3≥1]|F2
]X=0
= Q [X3 = 1 |HT,HT ]
= Q [HHH |HT, TH] =Q [HHH]
Q [HT, TH]
=8/27
4/9=
2
326 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Simple Example Continued
EP[1[X3≥1]ζ3|F2
]Node Expectation X2 ProbHH 1
2· 1 ·
(43
)3+ 1
2· 1 ·
(43
)2 (23
)2(43
)2HT 1
2· 1 ·
(43
)2 (23
)+ 1
2· 0 ·
(43
) (23
)2 1 12
(43
)2 23
TH 12· 1 ·
(43
)2 (23
)+ 1
2· 0 ·
(43
) (23
)2 1 12
(43
)2 23
TT 12· 0 ·
(43
) (23
)2+ 1
2· 0 ·
(23
)3 0 0
27 / 28
ConstructingBrownian
Motions andRadon-Nikodym
Derivatives
Steven R.Dunbar
StandardBrownianMotion
Binomial Trees
Using theRadon-Nikodymderivative
Simple Radon-Nikodym DerivativeExample Summarized
EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .
X2 EQ[1[X3≥1]|F2
]EP[1[X3≥1]ζ3|F2
]ζ2 Prob
2 1(43
)2 (43
)2 11 2
3
(12
) (43
)2 23
(43
) (23
)23
0 0 0(23
)2 0
28 / 28
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