Question: Non-symmetric random walk with stopping time

[mark_yao]

A random walk with a probability 0.7 being +1 and 0.3 being -1. Starting from 0
1. Walk 100 steps, what is the expectation at the end? ( easy )
2. Set a stop loss at -10, once the random walk hit -10 it stops there and use -10 as the final value. Walk 100 steps, what is the expectation?

Some friend of mine asked this, don't really know the answer for the second part. I guess it involves techniques like reflection?

 

[jovens]

Expectations are 40 for both question.
The process {X i-0.4*i} is martingale.

 

[mark_yao]

Expectations are 40 for both question.
The process {X i-0.4*i} is martingale.

Thanks for the reply. But I guess since we have a stopping time (\tau), that will change the problem a little bit. Let N = 100 and we have
(E[X_{\tau \land N}] - 0.4 E[\tau \land N] = 0)
This yields
( E[X_{\tau \land N}] = 0.4 E[\tau \land N]), since (\tau) has a positive probability to be smaller than N, the expectation will be smaller than 40.

The problem is can we find a easy way to find (E[\tau \land N]) ? I know there is some text book techniques but it is very complicated.

 

[leopardkk]

I think the key point is how to calculate p{ τ<=100}.
If we know this probability,the expectation will be(-10)*p+40*(1-p).
so i think we should first consider that probability of symmetric random walk. In that way,we can calculate the probability by reflection principle.My result is 1-p{x_100=10}-p{x_100=8}-p{x_100=6}-p{x_100=4}-p{x_100=2}-p{x_100=0}-p{x_100=-2}-p{x_100=-4}-p{x_100=-6}-p{x_100=-8}(all of this are under the symmetric probability measure). Then we transfer the symmetric probability measure to the real probability measure one by one by scaling the process(0.5/0.7)^#of heads*(0.5/0.3)^# of tails

 

[mark_yao]

I think the key point is how to calculate p{ τ<=100}.
If we know this probability,the expectation will be(-10)*p+40*(1-p).
so i think we should first consider that probability of symmetric random walk. In that way,we can calculate the probability by reflection principle.My result is 1-p{x_100=10}-p{x_100=8}-p{x_100=6}-p{x_100=4}-p{x_100=2}-p{x_100=0}-p{x_100=-2}-p{x_100=-4}-p{x_100=-6}-p{x_100=-8}(all of this are under the symmetric probability measure). Then we transfer the symmetric probability measure to the real probability measure one by one by scaling the process(0.5/0.7)^#of heads*(0.5/0.3)^# of tails

Thanks! I think this is what I am looking for. Can you refer me some text on the change of measure of discrete random walk? (I only saw the one for continuous process in Shreve II)

 

[snowave]

Hi, what about if the steps are not symmetric, i.e. left -1, right +2. How to solve this by using reflection principle? Thanks