Brownian Motion Recall the random walk { S n } n 0 under a - - PowerPoint PPT Presentation

Brownian Motion

• Recall the random walk {Sn}n≥0 under a probability measure

P: S0 = 0, and Sn =

n

• i=1

ξi, n ≥ 1, where ξi =

  

+1 with probability p; −1 with probability q = 1 − p and the {ξi}i≥1 are independent.

• Then Sn = 2Xn − n where Xn ∼ Bin(n, p), the binomial dis-

tribution.

1

• Results from the binomial distribution show that

E[Sn] = n(p − q) = 2n

• p − 1

2

• and

var(Sn) = 4npq.

• In the symmetric case, p = q = 1

2, and

E[Sn] = 0, var(Sn) = E

• S2

n

• = n.
• More generally,

cov(Sm, Sn) = m ∧ n.

2

• In fact, {Sn}n≥0 is a P-martingale (with respect to the natural

filtration).

• If 0 ≤ i ≤ j ≤ k ≤ l, then

Sj − Si =

j

• m=i+1

ξm and Sl − Sk =

l

• m=k+1

ξm are independent.

3

• Also, if j −i = l−k = m, say, then Sj −Si and Sl −Sk are both

sums of m ξ’s, and hence both have the same distribution as Sm.

• So, under P, {Sn}n≥0 has stationary and independent incre-

ments.

4

• Now for t ∈ R, t > 0, write

X(n)

t

= S⌊nt⌋ √n , where ⌊nt⌋ is the integer part of nt.

• Then E
• X(n)

t

• = 0, and

cov

• X(n)

s

, X(n)

t

• = 1

n⌊ns⌋ ∧ ⌊nt⌋ ≈ s ∧ t.

• By the Central Limit Theorem, if n is large then X(n)

t

is approximately N(0, t).

5

• Brownian motion is defined as the stochastic process with

these limiting characteristics.

• Specifically, a real-valued stochastic process {Wt}t≥0 is a P-

Brownian motion (or a P-Wiener process) if for some σ > 0: – for each s ≥ 0 and t > 0, Ws+t − Ws ∼ N(0, σ2t); – for each n ≥ 1 and times 0 ≤ t0 ≤ · · · ≤ tn, the increments {Wtr − Wtr−1} are independent; – W0 = 0; – Wt is continuous in t ≥ 0.

6

• If σ = 1, {Wt}t≥0 is standard Brownian motion.
• Notes:

– continuity (with P-probability 1) is in fact a consequence

• f the first two parts of the definition.

– the first two parts mean that {Wt}t≥0, like the random walk, has stationary and independent increments.

7

• A process with stationary and independent increments is

called a L´ evy process. – Brownian motion is the L´ evy process with Gaussian (nor- mally distributed) increments. – Brownian motion is the only L´ evy process with almost surely continuous sample paths. – Another L´ evy process is the counting process {Nt}t≥0 of a Poisson point process.

8

• Other properties of Brownian motion:

– Although almost surely continuous, it is almost surely nowhere differentiable; – Brownian motion will, with P-probability 1, eventually reach any real value, no matter how large, and will return to 0, also with P-probability 1; – Once Brownian motion hits a value, it immediately hits again infinitely often; – Brownian motion is self-similar.

9