Not that this changes the concept at all. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Theorem: Brownian Motion is nowhere-differentiable almost surely.
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At first sight, it might be thought that this is all that is needed. By countable additivity of probability measures, we can only infer the joint properties of the random variables at countable sets of times and not simultaneously on the uncountable set of nonnegative real numbers.
The best that can be done is to say that there is a continuous modification of the process. That is, if satisfies the first two properties above then there is a process with continuous sample paths and such that at each time, with probability one.
The third property above says that we always use such a continuous modification. Then, the properties of the sample paths, such as differentiability, are measurable events in the probability space with well defined probabilities. In fact, it can be shown that all continuous processes with independent increments are normally distributed and, therefore, can be expressed as a standard Brownian motion with a rescaling of the time axis plus a deterministic function.
Also, as a consequence of the central limit theorem , Brownian motion occurs as an approximation to discrete processes such as random walks in the limit as the step sizes go to zero.
So, Brownian motion is a very general process in stochastic process theory. As Brownian motion is a random process, any property of its sample paths is satisfied with a certain probability. The properties described below are meant in the almost sure sense.
That is, with probability one. Depending on how the probability space is constructed there could well be outcomes in which some or all of these properties are not true, but such outcomes will only occur with zero probability. Although it is continuous, the sample paths of Brownian motion are nowhere differentiable.
Historically, the first published example of a continuous but nowhere differentiable function was the Weierstrass function , in In fact, at a cursory glance, it does look very similar to Brownian motion. Previously, it was generally assumed that all functions under consideration were differentiable except, possibly, at some small set of nonsingular points. Most mathematicians believed this to be true for all continuous functions according to here.
The Weierstrass function was published as a pathological counterexample, showing that such functions do exist and is classed as a Monster of Real Analysis. In light of statements such as that by Hermite at the top of this post it is interesting that, in the form of Brownian motion, such functions are central to much mathematics developed in the 20th century with many important and practical applications.
If is a Brownian motion then, for any constant , it follows from the definition that the rescaled process is also a Brownian motion. This means that if we zoom in on the sample path of then, at every order of magnification, it still looks like a sample path of standard Brownian motion an applet demonstrating this property is available here.
This is in contrast to differentiable functions which become closer to a straight line when zoomed in. Brownian motion has infinite variation over all nontrivial intervals.
For finite variation processes, Riemann-Stieltjes integration can be used to make sense of terms like. Unfortunately, this does not work for Brownian motion because this integration is only well-defined for finite variation processes. In fact, it has been proven by Lebesgue that any function with finite variation is differentiable almost everywhere. The infinite variation property of Brownian motion is thus a consequence of nowhere differentiability.
Asked 2 years, 8 months ago. Active 2 years, 8 months ago. Viewed 1k times. Ile Ile 4 4 silver badges 12 12 bronze badges. Continuity does not imply differentiability. Add a comment. Active Oldest Votes. Neal Neal 30k 2 2 gold badges 61 61 silver badges bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
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