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Brownian Motion Continuity: Definition Versus Proof

Article Quant Q&A · Author: Nikolai Kl
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Summary

The document asks whether the standard Wiener process has almost surely continuous paths as a consequence of independent increments. It distinguishes mean-square continuity, which can be established from the distribution and independence of increments, from almost-sure path continuity, which is a stronger property and requires a more involved argument if it is not assumed at the outset.

The answers explain that many standard definitions include continuity as one of Brownian motion’s defining properties, alongside independent increments and normally distributed increments with variance equal to elapsed time. Other constructions can begin from different assumptions and establish continuity as a result; examples mentioned include Lévy-style constructions and representations using Haar wavelets or Fourier series. The discussion is conceptual rather than a full proof, and it does not provide the details of any construction. Its main lesson is to check which definition is being used before trying to derive path continuity from increment independence alone.

Key ideas

  • Independent increments and increment distributions establish mean-square continuity, but that does not by itself show almost-sure path continuity.
  • Many standard definitions take continuous paths to be a defining property of Brownian motion.
  • Alternative constructions can derive continuity, but require more advanced arguments.
  • The document names Lévy, Haar wavelet, and Fourier series approaches without proving them.

Tags

Statistics
Full text
# Proof: Brownian Motion Path Continious with Probability One


# Proof: Brownian Motion Path Continious with Probability One












How can one show that the paths of the standard Wiener process are continuous in $T$ with probability one? Can we just proof it with the assumption of independence ? Thank You in advance!

## Answer by siou0107 (score 6, accepted)

https://quant.stackexchange.com/a/61464

Using the distribution and independence of increments allows to prove $L^2$ (mean-square) continuity. Proving the a.s. continuity is much harder. Paul Lévy's construction of Brownian motion is related in Le Gall; an alternative is to construct the Brownian motion through Haar wavelet functions or Fourier series.

## Answer by oliversm (score 3)

https://quant.stackexchange.com/a/61466

#### It's part of the definition

I'd just like to re-iterate the comment by Kevin, (which as far as I can tell is the answer). There are three properties which define a standard Brownian motion / Wiener process:

- Independent increments.

- Normally distributed with variance equal to the time increment.

- The path is continuous.

Which hopefully any "standard" textbook on stochastics will re-iterate (Klebaner, Kloeden and Platen, Shreve, Oksendal, etc.).

However, as remarked in this comment, it is possible to drop this assumption and start with alternative constructions/definitions, from which continuity might be a consequence rather than a postulate. However, I suspect this is both more advanced, more nuanced, and less standard, so I don't know any references for this starting point.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.