Chapter 2

Question 1

Define stochatic process or random process

Answer 1

Collection of Random variables X(t)defined on the common probability space , indexed by elements of parameter set T for time. The set of all possible values for X is called the state space for T
X is a fucntion in 2 variables, t and omega

Question 2

Explain the trageory of a random process

Answer 2

The realisation or sample function/ path or the random process. For a fixed value of omega the function X(t) for all t is the trajectory

Question 3

Give an example of a stochastic process with a continuous state space and continuous time

Answer 3

Euro/ dollar exchange rate over time

Question 4

Give an example of a stochastic process with a continuous state space and discrete time

Answer 4

Annual inflation rates

Question 5

Give an example of a stochastic process with a discrete state space and discrete time

Answer 5

Number of earthquakes over a year

Question 6

If two variables X and Y are independent what can be said about their conditional expectation

Answer 6

The Expectation of X|Y is just expectation of X is X and Y are independent

Question 7

When do RVs have the same probability distribution

Answer 7

IFF for any bounded measurable function expectation of f(x) is the same as the expectation of F(y) then we write X==^d Y

Question 8

When is a stochastic process said to be a white noise

Answer 8

X(t) is a white noise if all X(t)s are iid we use notation Xn

Question 9

How do we know when Xn white noise is symmetric

Answer 9

The distribution of Xn equals the distribution of -Xn

Question 10

What are increments of steps of a random walk

Answer 10

Xi terms for all n that sum up to get the random walk

Question 11

If a random walk is not a symmetric random walk what can it be called?

Answer 11

Biased

Question 12

What does Sn/n tend to as n tends to infinity when Sn is a random walk

Answer 12

It tends towards E(xi) = mew because of law of large numbers

Question 13

Why is the central limit theorem a stronger result than the law of large numbers?

Answer 13

It is dealing with the converge of the distribution not just a central value

Question 14

Describe the gamblers ruin game set up

Answer 14

Each player has a starting capital. If A RV X is 1 - player s wins 1 unit if X is -1 - player f wins one unit. Game continues until one party loses all their capital.
Sn = x1+x2+… models amount won/lost by a player.

Question 15

Define p

Answer 15

Probability Xt = 1

Question 16

Define q

Answer 16

Probability Xt = -1

Question 17

Define a stationary process

Answer 17

A process X is stationary if for any t the joint distribution of the random vectors: (X(t1),X(t2)….X(tk)) and (X(t1+t),X(t2+t)….X(tk+t)) are identical. This implies Xr and Xs have the same distirbution and so therefore the expectation and variance of Xt is constant, not dependent on t.

Question 18

Is a random walk stationary?

Answer 18

No except in the degenerate case when Xi=0

Question 19

Is white noise stationary process?

Answer 19

Yes as all XN’s are iid

Question 20

What is the difference between the weakly stationary and strong stationarity definitions

Answer 20

It doesn’t say anything about the distributions

Question 21

Does White noise have independent increments?

Answer 21

No except where the Xns are not random in degenerate case - Increment rvs are dependent

Question 22

Does random walk process have independent increments?

Answer 22

Yes S1-S0=X1, S2-S1=X2 … etc are independent

Question 23

Explain the Markov property

Answer 23

A sequence of RVs Zn have the markov property if for any n:
P(Zn+1=z|Z0=z0,…,Zn=zn) = P(Zn+1=z|Zn=zn) . The next RV int he sequence is only dependent on the most recent value in the sequence.

Question 24

Does Random walk have a markov property?

Answer 24

Yes

Question 25

If a process has the markov property what does this mean if we want to find a probability k steps ahead of us?

Answer 25

We only have to condition on the most recent stage of the sequence

Question 26

Is white noise a martingale?

Answer 26

White noise is not a martingale it is a martingale only in degenerate case where Xn is a constant

Question 27

Is a random walk a martingale?

Answer 27

No unless it is an unbiased random walk!

Question 28

Define an increasing random process

Answer 28

Process that for any W X(t) is an increasing function: ie. all trajectories are increasing

Question 29

Define a continuous random process

Answer 29

Process that for any w, X(t) is a continuous function: ie. all trajectories are continuous