a**m
发帖数: 102 | 1 volume 1, chapter 5, exercise 5.5 (ii): compute the joint distribution of (M
,S) for an asymmetric random walk, where S_n is the position after first n
steps, M_n is the maximum of first n S_k 's. |
p*****k
发帖数: 318 | 2 not sure if anything tricky here: once you get (i),
simply replace (1/2)^n with p^[(n+b)/2] * q^[(n-b)/2]
there is a solution collection posted by QL365:
http://www.mitbbs.com/article_t/Quant/31186417.html |
a**m
发帖数: 102 | 3 Do you mean that we could still use reflection principle even in the
asymmetric case?
【在 p*****k 的大作中提到】
: not sure if anything tricky here: once you get (i),
: simply replace (1/2)^n with p^[(n+b)/2] * q^[(n-b)/2]
: there is a solution collection posted by QL365:
: http://www.mitbbs.com/article_t/Quant/31186417.html
|
p*****k
发帖数: 318 | 4 yes. reflection principle, from my understanding, is more of
a combinatorical statement, i.e., the number of paths which
satisfy certain conditions.
it's a different story if the step size is asymmetric |
a**m
发帖数: 102 | 5 Here the step size is always 1. But my understanding is that you need the
symmetry of the probability of up and down for every step to use the reflection principle,
since the reflection needs to promise that the new path realizes as the same probability as the old one does.
Please point out anything I am wrong about. Thanks.
【在 p*****k 的大作中提到】
: yes. reflection principle, from my understanding, is more of
: a combinatorical statement, i.e., the number of paths which
: satisfy certain conditions.
: it's a different story if the step size is asymmetric
|
p*****k
发帖数: 318 | 6 there were typos in my original post which i have corrected,
and i added the link to QL365's post. |
a**m
发帖数: 102 | 7 cool!
【在 p*****k 的大作中提到】
: there were typos in my original post which i have corrected,
: and i added the link to QL365's post.
|
