Admistrivia

• Read section 8.2 and 8.3

Models of Brownian motion

Boundaries

• Exact Brownian motion doesn't exist (velocities would be infinite for example.)
• It is a very good approximation though
• What might boundaries look like for a random walk?

• No boundaries (state space -infinity to +infinity)
• absorbing boundary (i.e. gamblers ruin)
• Reflecting boundary (i.e. birth-death with imigration)

Reflecting boundary

• E(R(t)|R(0)=0) = sqrt(2t/pi)
• p(y,t|x) = phi_t(y-x) + phi_t(-y-x)

Absorbing boundary

• Start at B(0) = x.
• Recall tau, our first hitting time of B(t) = 0
• Let A(t) = B(t) up to time tau
• Let A(t) = 0 after tau.

• NOTE: I'll use infinitesimal approach. Book uses CDF approach.
• Use reflection principle to determine P(min X < 0 and y < X(t) < y+h|X(0) = x)
  • Draw picture
  • Same probability as -y > X(t) > -y-h.
  • so answer is phi_t(-x-y)h
• Use law of total probability to compute: P(min X > 0 and y < X(t) < y+h|X(0) = x)
  • P(y < X(t) < y+h|X(0) = x) = phi_t(y-x)h
  • answer = (phi_t(y-x) - phi_t(-y-x))h
• Density = limit

Brownian Bridge

• aka tied down Brownian motion
• Condition on X(0) = 0, and X(1) = 0.
• E(X(t)) = 0.
• for 0 < s < t < 1: Cov(X(s),X(t)) = s(1-t).
• s = t: Var(X(s)) = s(1-s).
• E(X(t)|X(s)) = E(X(t)- beta X(s)+ beta X(s)hi

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.