L evys Construction L evy showed that the definition of Brownian - - PowerPoint PPT Presentation

L´ evy’s Construction

• L´

evy showed that the definition of Brownian motion {Wt}t≥0 is internally consistent, by constructing a process with the required properties, including continuity of the sample paths.

• The constructed process {Xt}t∈[0,1] is the limit of piecewise

linear, continuous approximations

• X(n)

t

• t∈[0,1]

.

1

• The construction is based on the properties of the Brownian

Bridge: – If {Wt}t∈[0,1] is standard Brownian motion and 0 ≤ t1 < t < t2 ≤ 1, then conditionally on Wt1 = a and Wt2 = b, Wt ∼ N

• a + t − t1

t2 − t1 (b − a), (t2 − t)(t − t1) t2 − t1

• .
• In particular, at the midpoint t = (t1 + t2)/2,

Wt ∼ N

a + b

2 , t2 − t1 4

• .

2

• The construction uses a countable collection of independent

N(0, 1) variables {ξt}t∈D. – Here D is the set of dyadic fractions in (0, 1]; that is, points of the form

k 2n, k ∈ {1, 2, . . . , 2n}, n ∈ N.

– Each dyadic fraction has a unique finite binary expansion.

3

• The first approximation is

X(0)

t

= tξ1, t ∈ [0, 1]. – This gives the correct distribution for X1 ∼ N(0, 1) (and

• f course X0 = 0).
• The second approximation X(1)

t

keeps these values: X(1)

t

= X(0)

t

, t ∈ {0, 1}; adjusts the midpoint: X(1)

t

= X(0)

t

+ 1 2ξt, t = 1 2; and linearly interpolates these points for other t.

4

• The second approximation gives the correct joint distribution

for X1

2 and X1.

• The third approximation X(2)

t

keeps these values: X(2)

t

= X(1)

t

, t ∈

• 0, 1

2, 1

• ,

adjusts the midpoints t = 1

4 and t = 3 4,

X(2)

t

= X(1)

t

+ 1 2 √ 2ξt, t ∈

1

4, 3 4

• ;

and again linearly interpolates these points for other t.

5

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t x0

First approximation 6

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t x0

First two approximations 7

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t x0

First three approximations 8

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t x0

First four approximations 9

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t x0

Final process 10

• L´

evy showed that there exists a limit process {Xt}t∈[0,1] such that P

• X(n)

t

→ Xt as n → ∞, uniformly for t ∈ [0, 1]

• = 1.
• Because each
• X(n)

t

• t∈[0,1]

is continuous and the conver- gence is uniform, P[Xt is continuous on [0, 1]] = 1.

• Each
• X(n)

t

• t∈[0,1]

has the correct distribution at t = k

2n, k ∈

{0, 1, . . . , 2n}, so {Xt}t∈[0,1] has the correct distribution at all dyadic points, and by continuity at all t ∈ [0, 1].

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Some Brownian Motion Tricks

• No memory: if {Wt}t≥0 is Brownian motion and s ≥ 0 is a

fixed time, then {Ws+t − Ws}t≥0 is also Brownian motion, and it is independent of {Wt}0≤t≤s.

• The same is true for a random time T, provided it is a stop-

ping time.

• T is a stopping time for {Wt}t≥0 if, for each t, the event {T ≤

t} is Ft-measurable, where {Ft}t≥0 is the natural filtration of {Wt}t≥0.

12

• Hitting time for a level a:

Ta = inf{t ≥ 0 : Wt = a}. – A hitting time is a stopping time: {Ta ≤ t} = {Ws = a for some s, 0 ≤ s ≤ t} ∈ Ft.

• Distribution of a hitting time:

P[Wt > a] = P[Ta < t ∩ Wt − WTa > 0] = P[Ta < t] × P[Wt − WTa > 0|Ta < t] = 1 2P[Ta < t]. – So P[Ta < t] = 2P[Wt > a] = 2

• 1 − Φ
• a/

√ t

• .

13

• Reflection Principle: {Wt}t≥0 is Brownian motion and T is a

stopping time. Let ˜ Wt =

  

Wt, t ≤ T; 2WT − Wt, t > T. – That is, for t > T, ˜ Wt is Wt, reflected around the level WT.

• Then { ˜

Wt}t≥0 is also Brownian motion.

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• Maximum over an interval: if Mt = max0≤s≤t Ws, then for

a > 0 and x ≤ a, P[Mt ≥ a ∩ Wt ≤ x] = 1 − Φ

• 2a − x

√ t

• .
• Proof:

P[Mt ≥ a ∩ Wt ≤ x] = P[Ta ≤ t ∩ Wt ≤ x] = P[Ta ≤ t ∩ 2a − x ≤ ˜ Wt] = P[2a − x ≤ ˜ Wt] = 1 − Φ

• 2a − x

√ t

• .
• Used in pricing barrier options.

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t w

Reflection; t = 1 , a = 1 , x = 0.8 16

• Transforming Brownian motion: if {Wt}t≥0 is standard Brow-

nian motion, then so are

• 1. {cWt/c2}t≥0 for any real c = 0;
• 2. {tW1/t}t≥0, where tW1/t is taken to be 0 when t = 0;
• 3. {Ws − Ws−t}0≤t≤s for any fixed s ≥ 0.
• Case 1 is self-similarity: no characteristic time scale.
• Case 2 shows that if we have constructed {Wt}0≤t≤1, we can

extend the construction to {t ≥ 1}, and hence to {t ≥ 0}.

• Case 3 is time-reversibility.

17