c*********s
发帖数: 327 | 1 Suppose that x is a Brownian motion with no drift and unit variance, i.e. dx
= dz. If x starts at 0, what is the probability that x hits 3 before
hitting -5?
答案是： 5/8吗？ | c*********s
发帖数: 327 | 2 然后这个怎么作？
what if the drift is m, i.e.
dx = m dt + dz?
dx
【在 c*********s 的大作中提到】
: Suppose that x is a Brownian motion with no drift and unit variance, i.e. dx
: = dz. If x starts at 0, what is the probability that x hits 3 before
: hitting -5?
: 答案是： 5/8吗？
| a***s
发帖数: 616 | 3 Yes. Gambler's Ruin.
For the second one, the idea is the same: find a martingale.
One candidate is that e^{-2*m*X_t}, where X_t = m*t + B_t with B_t a
standard Brownian Motion. By Ito's formula, this is a martingale.
Then by standard argument for Gambler's Ruin, you can get the result.
The general solution for hitting a before b is
( 1 - e^{-2*m*b} ) / ( e^{-2*m*a} - e^{-2*m*b} ).
Here the BM is assumed to start in the interval [a,b] (or [b,a], does not
matter whic is larger).
dx
【在 c*********s 的大作中提到】
: Suppose that x is a Brownian motion with no drift and unit variance, i.e. dx
: = dz. If x starts at 0, what is the probability that x hits 3 before
: hitting -5?
: 答案是： 5/8吗？
| h**f
发帖数: 149 | 4 ( 1 - e^{-2*m*b} ) / ( e^{-2*m*a} - e^{-2*m*b} ).
I got a different result from another document, isnt it
( 1 + e^{-2*m*b} ) / ( e^{-2*m*a} + e^{-2*m*m} ).
?
Please confirm that if anyone knows the answer. |
|
