Chapter 7 - Brownian Motion and Martingales

Question 1

What is a stochastic process?

Answer 1

A stochastic process is a sequence of values of some quantity where the future values cannot be predicted with certainty.

Question 2

What is the definition of Standard Brownian Motion (Wiener Process)?

Answer 2

A stochastic process Wt is a Wiener Process if:
1. W0 = 0
2. Wt has continuous sample paths
3. For t > u >= 0
Wt-Wu ~N(0, t-u)
4. Independent increments
Wt-Ws is independent of Wu for u<s

Question 3

What is a martingale?

Answer 3

The exp

Question 4

Explain what a Markov Process is.

Answer 4

A Markov process is one where If we know the latest value of the process, we have all the information required to determine the probabilities for future values. i.e. having the whole historical values would not help us any further

Question 5

Why is a Weiner Process Marvok?

Answer 5

A Weiner process is Markov as the increments are independent, so we only need to know the current point to determine probabilities for the future. Having the historical paths will not help us any further.

Question 6

What is the formula of a Brownian Motion Process constructed out of a SBM?

Answer 6

Zt = Z0 + mut + sigmaWt

mu - drift parameter
sigma - diffusion parameter (volatility)
Wt - Standard Brownian Motion (Wiener Process)

Question 7

What is the definition of Brownian Motion Process?

Answer 7

2. Wt has continuous sample paths
3. For t > u >= 0
   Wt-Wu ~N(mu(t-u) , sigma^2(t-u)
4. Independent increments
   Wt-Ws is independent of Wu for u<s

Question 8

What is the Covariance of a Wiener Provess?

Answer 8

Cov(Ws,Wt) = min(s,t)

Proof:
Cov(Ws + Wt - Ws, Ws) = Cov(Ws,Ws) + Cov(Wt-Ws,Ws) = var(Ws) + 0 = s

Question 9

Can we use the cov(Ws,Wt) = min(s,t) to prove a process is a Weiner Process?

Answer 9

Yes

Question 10

What is the formula for a Scaled Weiner Process?

Answer 10

Xt = sqrt(c)*W_(t/c)

i.e. time has been slowed down if C>1 and sped up is c<1

Question 11

What is the formula for a Scaled Weiner Process?

Answer 11

Xt = t*W_(1/t)

Question 12

What is the formula for a Correlated Weiner Process?

Answer 12

Zt = pWt1 + sqrt(1-p^2)Wt2

Wt1 and Wt and independent Weiner processes
p is the correlation coefficient between Zt & WWt1

Question 13

What are the downfalls to using Brownian Motion to describing movement of the market indices?

Answer 13

• Good in the short run
• Bad as certain to become negative in the long run
• and daily movements off 100 will occur just as frequently when the process is at level 100 or at level 10,000.

Question 14

What is the Geometric formula for a Geometric Brownian Motion Process?

Answer 14

St = exp(Zt)
where Zt=Z0 + mut + sigmaWt (the brownian motion process)

Question 15

A martingale is a stochastic process such that:

Answer 15

For a stochastic process Xt, s<t

E[Xt|Fs] = Xs
E|Xt|< inf for all t

In words, the expected value of the process at time t give we are at time s and we know all the information up to and including S

Question 16

What is the formula to show a stochastic process Xt is a submartingale?

Answer 16

W[Xt|Xs] >= Xs

Question 17

What is the formula to show a stochastic process Xt is a supermartingale?

Answer 17

W[Xt|Xs] < Xs

Question 18

What other properties do we know of a Weiner Process?

Answer 18

• All Wiener processes are a martingale
• All Wiener processes are Markov (due to independent increments)
• Cov(ws,wt) = Min{(t,s)
• Sample paths are not differentiable anywhere