L´ evy’s Construction • L´ evy showed that the definition of Brownian motion { W t } t ≥ 0 is internally consistent, by constructing a process with the required properties, including continuity of the sample paths. • The constructed process { X t } t ∈ [0 , 1] is the limit of piecewise � � X ( n ) linear, continuous approximations . t t ∈ [0 , 1] 1
• The construction is based on the properties of the Brownian Bridge: – If { W t } t ∈ [0 , 1] is standard Brownian motion and 0 ≤ t 1 < t < t 2 ≤ 1, then conditionally on W t 1 = a and W t 2 = b , � � ( b − a ) , ( t 2 − t )( t − t 1 ) a + t − t 1 W t ∼ N . t 2 − t 1 t 2 − t 1 • In particular, at the midpoint t = ( t 1 + t 2 ) / 2, � a + b , t 2 − t 1 � W t ∼ N . 2 4 2
• The construction uses a countable collection of independent N (0 , 1) variables { ξ t } t ∈D . – Here D is the set of dyadic fractions in (0 , 1]; that is, k 2 n , k ∈ { 1 , 2 , . . . , 2 n } , n ∈ N . points of the form – Each dyadic fraction has a unique finite binary expansion. 3
• The first approximation is X (0) = tξ 1 , t ∈ [0 , 1] . t – This gives the correct distribution for X 1 ∼ N (0 , 1) (and of course X 0 = 0). • The second approximation X (1) keeps these values: t X (1) = X (0) , t ∈ { 0 , 1 } ; t t adjusts the midpoint: + 1 2 ξ t , t = 1 X (1) = X (0) 2; t t and linearly interpolates these points for other t . 4
• The second approximation gives the correct joint distribution for X 1 2 and X 1 . • The third approximation X (2) keeps these values: t 0 , 1 � � X (2) = X (1) , t ∈ 2 , 1 , t t adjusts the midpoints t = 1 4 and t = 3 4 , 1 � 1 4 , 3 � X (2) = X (1) + √ ; 2 ξ t , t ∈ t t 4 2 and again linearly interpolates these points for other t . 5
First approximation 1.0 0.8 0.6 x0 0.4 0.2 0.0 6 0.0 0.2 0.4 0.6 0.8 1.0 t
First two approximations 1.0 0.8 0.6 x0 0.4 0.2 0.0 7 0.0 0.2 0.4 0.6 0.8 1.0 t
First three approximations 1.0 0.8 0.6 x0 0.4 0.2 0.0 8 0.0 0.2 0.4 0.6 0.8 1.0 t
First four approximations 1.0 0.8 0.6 x0 0.4 0.2 0.0 9 0.0 0.2 0.4 0.6 0.8 1.0 t
Final process 1.0 0.8 0.6 x0 0.4 0.2 0.0 10 0.0 0.2 0.4 0.6 0.8 1.0 t
• L´ evy showed that there exists a limit process { X t } t ∈ [0 , 1] such that � � X ( n ) → X t as n → ∞ , uniformly for t ∈ [0 , 1] = 1 . P t � � X ( n ) • Because each is continuous and the conver- t t ∈ [0 , 1] gence is uniform, P [ X t is continuous on [0 , 1]] = 1 . � � X ( n ) has the correct distribution at t = k • Each 2 n , k ∈ t t ∈ [0 , 1] { 0 , 1 , . . . , 2 n } , so { X t } t ∈ [0 , 1] has the correct distribution at all dyadic points, and by continuity at all t ∈ [0 , 1]. 11
Some Brownian Motion Tricks • No memory: if { W t } t ≥ 0 is Brownian motion and s ≥ 0 is a fixed time, then { W s + t − W s } t ≥ 0 is also Brownian motion, and it is independent of { W t } 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 { W t } t ≥ 0 if, for each t , the event { T ≤ t } is F t -measurable, where {F t } t ≥ 0 is the natural filtration of { W t } t ≥ 0 . 12
• Hitting time for a level a : T a = inf { t ≥ 0 : W t = a } . – A hitting time is a stopping time: { T a ≤ t } = { W s = a for some s , 0 ≤ s ≤ t } ∈ F t . • Distribution of a hitting time: P [ W t > a ] = P [ T a < t ∩ W t − W T a > 0] = P [ T a < t ] × P [ W t − W T a > 0 | T a < t ] = 1 2 P [ T a < t ] . – So √ � � �� P [ T a < t ] = 2 P [ W t > a ] = 2 1 − Φ a/ t . 13
• Reflection Principle: { W t } t ≥ 0 is Brownian motion and T is a stopping time. Let W t , t ≤ T ; ˜ W t = 2 W T − W t , t > T. – That is, for t > T , ˜ W t is W t , reflected around the level W T . • Then { ˜ W t } t ≥ 0 is also Brownian motion. 14
• Maximum over an interval: if M t = max 0 ≤ s ≤ t W s , then for a > 0 and x ≤ a , � � 2 a − x P [ M t ≥ a ∩ W t ≤ x ] = 1 − Φ √ . t • Proof: P [ M t ≥ a ∩ W t ≤ x ] = P [ T a ≤ t ∩ W t ≤ x ] = P [ T a ≤ t ∩ 2 a − x ≤ ˜ W t ] = P [2 a − x ≤ ˜ W t ] � � 2 a − x = 1 − Φ √ . t • Used in pricing barrier options. 15
Reflection; t = 1 , a = 1 , x = 0.8 1.2 1.0 0.8 w 0.6 0.4 0.2 0.0 16 0.0 0.2 0.4 0.6 0.8 1.0 t
• Transforming Brownian motion: if { W t } t ≥ 0 is standard Brow- nian motion, then so are 1. { cW t/c 2 } t ≥ 0 for any real c � = 0; 2. { tW 1 /t } t ≥ 0 , where tW 1 /t is taken to be 0 when t = 0; 3. { W s − W s − 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 { W t } 0 ≤ t ≤ 1 , we can extend the construction to { t ≥ 1 } , and hence to { t ≥ 0 } . • Case 3 is time-reversibility . 17
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