Admistrivia

Optional: read 6.6
Read section 8.1

Brownian motion

History

1827: Robert Brown watch grains of pollen

1905: Einstein described why--its due to atoms.

1927: Norbert Wiener gave mathematical model

Aside: 1900, Louis Bachelier used brownian motion to describe the stock market. Resoundingly ignored! Shows--don't be too far ahead of your friends if you want recgonization.

Constructing a Brownian motion stochastic process

Two good examples of infinitely divisable processes:

Poisson
Normal

Lets use this to create a Brownian motion process

Discrete time Normal process is easy: t = 0,1,2,3,4,...
Want to create more refined process for t = 0,.5,1,1.5,2,...
A = (X_{t + .5} - X_t) is independent of B = (X_{t
+ 1} - X_t+.5)
C = A + B = (X_{t + 1} - X_t), so we "know" it
Define D = A - B
Key fact: C and D are independent (check!)
E(C|C) = C, E(D|C) = 0, Var(C|C) = 0, Var(D|C) = 1
Thus: E(A|C) = C/2 and Var(A|C) = 1/2.
Can simulate as accurately as we like

Details: involving PDEs!

Define: p(y,t|x) = density of B(s+t)=y given B(s) = x.

P(y,t|x)&ge zero. (Duh!)
intergral = 1.
limit t --> 0 is zero if y &ne 0.
Einstien: dp/dt = 1/2 sigma² d²p/dx².
Solution: p(y,t|x) = Normal(mean = x, var = t).
It solves equations and is the only solution (difficult pdf result)

Dean P. Foster

Last modified: Wed Apr 14 10:43:18 EDT 2004