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!
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$\begingroup$ What have you tried to solve this? $\endgroup$– Bob Jansen ♦Mar 1, 2021 at 12:36
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$\begingroup$ I applied Bayes law to $P(W_t | W_{t-1} + W_{t-2} + ... + W_{t-n})$, where $t > n$. Then I arranged the probability which should be equal to one as the past realisations should be independent from future observations. $\endgroup$– Nikolai KlMar 1, 2021 at 12:39
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1$\begingroup$ What do you mean with ''prove'' this? It's part of the definition. A Brownian motion has almost surely continuous paths, i.e. the probability of getting a discontinuous path is zero. That's part of the usual definition. You can't ''prove'' that the multiplication in a group is associative either. It's part of its definition. $\endgroup$– KevinMar 1, 2021 at 12:46
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$\begingroup$ Thas already an insight. My mathematical background is not that strong but I in class were given the statement 'That one can proof that the path of a brownian Motion is continous with the probability of one'. I thought I could solve it using induction. $\endgroup$– Nikolai KlMar 1, 2021 at 12:59
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3$\begingroup$ If you take of BM definition as "The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments." (from Wikipedia) I think proving continuity is of interest actually. $\endgroup$– Bob Jansen ♦Mar 1, 2021 at 15:23
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2 Answers
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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.
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$\begingroup$ In the end I am looking for that proof! Ill have a look at the book, thank You! $\endgroup$ Mar 1, 2021 at 18:12
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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.
