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发帖数: 87 | 1 Suppose that x is a Brownian motion with drift m and unit variance, i.e. dx
=m dt + dz. If x starts at 0, what is the probability that x hits 3 before
hitting -5? | m**********s
发帖数: 87 | | j***e
发帖数: 72 | 3 This is a very standard question, post your answer.
If I remember correct, the result has two cases, m=sigma^2/2 and m \neq sigm
a^2/2, here you gave sigma=1
dx
before
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| a**********n
发帖数: 5 | 4 Since X_t = m t + z_t.
First, multiply -2m on both sides, then take exponential on both sides. At
this point,
exp(-2m X_t) is a martingale, you can check it.
By optional sampling theorem,
you can get the answer:
p = (1- exp(10m) ) / (exp(-6m) - exp(10m)) , if m != 0
p = 5/8 if m = 0
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| m**********s
发帖数: 87 | 5 Thanks for your answer!! | e**********n
发帖数: 359 | 6 Fix a, so that exp(ax_t) is a martingale. There is a more general solution
to this kind of problem, check out any book on stochastic calculus. |
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