3 ms·

As long as you're nit picking, I'd suggest you re read carefully what you wrote, because it is wrong. What you meant to say is that on a set of measure zero, t

by mturmon 13y ago

As long as you're nit picking, I'd suggest you re read carefully what you wrote, because it is wrong. What you meant to say is that on a set of measure zero, the paths can have points where continuity is not satisfied. But you did not say that. In any case, the version of Brownian motion I studied was the version that excludes this null set. ;-)

yummyfajitas 13y ago
Yes I meant to say differentiable, brownian motion is always continuous. But with probability zero, brownian motion can generate a differentiable curve. Take the typical random walk construction as the limit of a piecewise continuous curve. I.e., 50% chance of being y=x, 50% chance of being y=-x on [0,1], then subdivide. With zero probability, the walk can converge to a straight line x=t, which is certainly differentiable. It's early in the morning and I had no caffeine.
- mturmon 13y ago
  You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not have this property; in fact, you could make a version (in the technical sense of that term, see http://almostsure.wordpress.com/2009/11/03/stochastic-processes-indistinguishability-and-modifications/ http://almostsure.wordpress.com/2009/11/03/stochastic-proces...) of the B.M. that uses your construction, but removes the null set of differentiable sample paths. In other words, this property ("differentiability on null sets") is not characteristic of a B.M., it is incidental to your construction. I can't believe you bothered to fuss about these hypothetical properties on null sets when we are dealing with cadlag processes on a separable index set. Sheesh.
  - LeFuneste 13y ago
    I agree, no one cares in probability theory on the behaviour of BM on this null set. It could be as well a Weierstrass function that it will not change anything to the law of your process.

by mturmon 13y ago

yummyfajitas 13y ago

Yes I meant to say differentiable, brownian motion is always continuous. But with probability zero, brownian motion can generate a differentiable curve. Take the typical random walk construction as the limit of a piecewise continuous curve. I.e., 50% chance of being y=x, 50% chance of being y=-x on [0,1], then subdivide. With zero probability, the walk can converge to a straight line x=t, which is certainly differentiable. It's early in the morning and I had no caffeine.

mturmon 13y ago
You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not have this property; in fact, you could make a version (in the technical sense of that term, see http://almostsure.wordpress.com/2009/11/03/stochastic-processes-indistinguishability-and-modifications/ http://almostsure.wordpress.com/2009/11/03/stochastic-proces...) of the B.M. that uses your construction, but removes the null set of differentiable sample paths. In other words, this property ("differentiability on null sets") is not characteristic of a B.M., it is incidental to your construction. I can't believe you bothered to fuss about these hypothetical properties on null sets when we are dealing with cadlag processes on a separable index set. Sheesh.
- LeFuneste 13y ago
  I agree, no one cares in probability theory on the behaviour of BM on this null set. It could be as well a Weierstrass function that it will not change anything to the law of your process.