Chapter 9 Brownian Motion And Martingales

Question 1

Standard Brownian Motion

Answer 1

SBM (aka Weiner Process) is a stochastic process {Wt, t>=0} with a state space S=R and has the following properties:

1) Wo =0
2) Wt has continuous sample paths.
3) For any 0<=s

Question 2

Properties of Standard Brownian Motion (10 points)

Answer 2

1) Wo=0
2) Wt has continuous sample paths.
3) For any 0<=s

Question 3

Brownian Motion

Answer 3

BM is a stochastic process {Zt, t>=0} with state space S=R that satisfies the following properties:

1) Zo is not necessarily 0.
2) Zt has continuous sample paths.
3) For any 0<=s

Question 4

Geometric Brownian Motion

Answer 4

{St, t>=0} is GBM and has the following properties.

1) St~logN(Zo+mu.t,sigma-squared.t)
2) St >=0

Question 5

Martingale

Answer 5

A martingale is a stochastic process for which its current share price is the optimal estimator of its future value.

Given a filtered probability space (Gamma, F, Ft, P) a stochastic process Xt is called a martingale with respect to the filtration Ft, if:

1) Xt is adapted to Ft
2) E[|Xt|]

Question 6

Levy’s Theorem

Answer 6

If a stochastic process has the property Cov(Xs, Xt)=min(s,t) then the process Xt follows SBM.