c**********e
发帖数: 2007 | 1 Let W^1_t, ..., W^n_d be independent Brownian motions. Define
R_t = sqrt(W^1_t*W^1_t ... + W^n_d*W^n_d).
What is dR_t? (The SDE for R_t) | z****i
发帖数: 406 | | c**********e
发帖数: 2007 | 3 The question asks to show d R_t = (n-1)/2 dt + dW_t
for some W_t.
But it seems to me it does not make sense. R_t should
not be a BM with drift. So the question is wrong?
【在 z****i 的大作中提到】
: Bessel process?
| p*****k
发帖数: 318 | 4 careerchange, the SDE should be:
dR_t = (n-1)/(2*R_t) dt + dW_t
the drift at R_t=0 is actually infinite.
not quite sure what the best way is to explain it, but if you are
familiar with Einstein's sqrt(t) result for diffusion, it might help.
or think about Polya's famous result that 3d random walk is transient,
i.e., the prob returning 0 is strictly less than 1.
the hard part for me was to recognize that
sum{n=1 to d} W_n *(d W_n)/R_t
is actually a brownian motion once you apply ito's lemma.
i re |
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