Admistrivia

• Read parts of section 8.4 (p 508-514 and 514-521. Stop each subsection when your eyes glaze over.)

More models of Brownian motion

BM with drift

• Suppose B(t) is the standard BM (i.e. no sigma so var(B(t)) = t)
• Drift: X(t) = mu t + sigma B(t)
• Drifts up at rate t.
• Theorem: Y(t) = exp(-2mu X(t)/sigma²) is a martingale.
• Called Girsanov's martingale
• Proof sketch:
  • E(Y(t+h)|Y(t)) = Y(t)exp(-2 mu (X(t+h)-X(t))/sigma²)
  • if E(exp(-2 mu (X(t+h)-X(t))/sigma²)) = 1 we have a martingale!
  • exp(delta) = 1 + delta + delta²/2 approximately
  • E(delta) = -2mu²h/sigma²
  • E(delta²) = 2mu²h/sigma²
  • QED
• From this we can compute the probability of going bankrupt.
• We can also compute the probability of the maximum.

Geometric BM

• Now let Y(t) = exp(X(t))
• Since constants aren't as above, not a martingale
• Good model for stock prices

Future directions for study and review of course

Markov chains

• Leads in to dynamical systems
• modeling physics, engineering, chemistry, DNA, and other "clean" settings

Poisson processes

• Leads into semi-martingales
• Modern model of Markets (allows for jumps, and difusions)
• dY = mu dt + sigma dN

BM

• Leads into SDE's
• Classical PDE.
• Classical finance.

Martingales

• Leads into measure theory
• Philosophy of knowledge (sigma-fields)
• dY = mu dt + sigma dM