Hey y'all,
I remember in our last meeting (when we were discussing the filter strategy and barrier strategies) that it was mentioned that since BMs have normally distributed increments, the value of a BM at t+1 can be anything from +inf to -inf (only in expectation is it BM(t) ).
In some sense its ridiculous to talk about t and t+1 since dt is infinitesimal. But is it the case that at each infinitesimal increment, the BM could concievably be hitting +inf or arbitrarily close and coming back to some "mean"? What happens when the BM over the next infinitesimal is arbitrarily large. Does the Martingale property now imply that it stays that large? Maybe I'm just being an idiot.
Is my understanding of this correct? Is there somewhere in Durret or Shreve I could look to figure this out (assuming I can understand Durret or Shreve)? Perhaps I can just add assorted facts about BMs to our list of things to discuss on Sunday.
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the distribution of B_{t+dt} - B_t is in some loose sense Normal(0,dt), so yes for arbitrary large M, there is a positive probability that B_{t+dt} - B_t > M, but that doesn't mean the BM every gets to infinity, it takes values on (-infty, infty). Also the sample path of a BM are almost surely continuous, so for any epsilon there is a delta such that P(|B_{t+delta} - B_t| < epsilon) = 1
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