Admistrivia

• Optional: read 6.6
• Read section 8.1

Brownian motion: 8.1 continued

Notation: little phi, big phi is normal distrubtion

Axioms

• increments are normal with mean =0, var = t sigma².
• increments are independent.
• B(0) = 0.
• (Difficult one) B(t) is continuous.

Typical Brownian motion: Chance of George II being elected

Covariance

• Cov(B(s),B(t)|B(T)) = Cov(B(s)-B(T),B(t)-B(T)|B(T))
• = Cov(B(s)-B(T),B(t)-B(T)+ B(s)-B(s)|B(T))
• = Cov(B(s)-B(T),B(s)-B(T)|B(T)) + Cov(B(s)-B(T),B(t)-B(s)|B(T))
• = Var(B(s)-B(T),B(s)-B(T)|B(T)) + 0
• = (s-T)sigma²

Invariance principle

• Let B(t) = sum(nt) X(i)/sqrt(n)
• Check assumptions:
  • increments are close to normal (for large n) via CLT
  • increments are independent by construction
  • B(0) = 0.
  • Continuous? Let the mathematicians worry about this one. (If we assume "2+epsilon" moments it should be continuous.)
• Hence all "sums" look the same!

Brownian motion: 8.2

Hitting time

• Well defined (because of continuity)
• may be infinite?

Reflection principle:

• B*(t) = x + (B(t)-x) for t &le tau_x
• B*(t) = x - (B(t)-x) for t > tau_x
• Note: B*(t) satisfies axioms of brownian motion
• same variance as before
• Draw picture (eg p 492)

Maximum of a brownian motion

• P(max B(s) > x) = P(tau_x < t)
• = P(B*(t) > x and B(t) < x) + P(B(t) > x and B*(t) < x)
• = P(B*(t) > x) + P(B(t) > x)
• = 2 P(B(t) > x)
• = 2(1 - Phi_t(x))
• = 2 Phi_t(-x)

Hitting time distribution

• P(tau_x < t) = P(max B(s) > x) = 2 Phi_t(-x)

Hitting time distribution

• P(zero between t and t+s) = 2/pi arctan sqrt(s/t)
• Proof via graphs!

Number of zeros

• P(zero in interval epsilon³ to epsilon) = 1 - O(epsilon)
• Expected number "near" zero is infinite
• Worse than that: Call a zero at t, "delta-natural" if there is another zero in (t,t+delta).
• Probability of being delta-natural is 1.
• Call a point a left-limit if it is natural for all delta.
• Probability of being left-limit is 1.
• If number of zeros is conuntable, probabilty of all being left-limits is 1.
• But, this means there can be no "last" zero.
• Hence there are an uncountable number of zeros.