# Talk:Wiener process

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I erased the reference to random walk, since it was imprecise and the information about the variance of the Wiener process was yet stated above. I added some words about the role of the Wiener process in Pure and Applied Mathematics. Gala.martin 20:27, 6 February 2006 (UTC)

The picture previously showed a Brownian bridge, not a general Wiener process. Although a simulation of a Wiener process might turn out to look like a Brownian bridge by pure chance, I think it is preferrable to use another picture here, so I changed it. --PeR 12:20, 30 June 2006 (UTC)

I think that we had better put

$R(s, t)=E[X(s)X(t)]=\sigma^2\min(s,t)$

covariance

$\gamma(s,t)=E[(X(s)-m(s))(X(t)-m(t))]=...$

and its derivative.

$a=t0

Limit of the Sum [W(tk)-W(t(k-1))]^2 = b-a

$(I)\int_{a}^{b} W(t)\,dW(t) = (W^2(b)-W^2(a))/2-(b-a)/2.$
$(S)\int_{a}^{b} W(t)\,dW(t) = (W^2(b)-W^2(a))/2.$

—Preceding unsigned comment added by Jackzhp (talkcontribs)

Is the definition sound? The third property is "W_t has independent increments with distribution [...]". Does it mean that with two such increments the property is enforced, or _all_ increments should have the given distribution? I think all increments should satisfy the property, but this is not what I understand from the sentence. The second condition (continuity) is unusual. I am more used to the definition provided there: [1] —Preceding unsigned comment added by 81.249.156.207 (talkcontribs)

The definition is sound. The continuity condition is not at all unusual, although many authors don't include continuity in their definition and instead show that every Brownian motion has an a.s. continuous "modification." –Joke 20:57, 3 January 2007 (UTC)

Why the stochastic differential equation whose solution is the Wiener process is not mentioned in the page? Albmont 18:41, 3 January 2007 (UTC)

Because that wouldn't make any sense. Since SDEs are most simply generated by a Wiener process, the corresponding equation is d(Wiener)=d(Wiener). –Joke 20:57, 3 January 2007 (UTC)

Ok, I didn't express myself correctly. I mean, if I have this SDE d(X) = d(Wiener), then the solution is not simply X = Wiener, but something like X(t) can be simulated by X(t) ~ X(0) + sqrt(t) N(0,1) (or something else; I am starting to get the feel of SDEs). How can I get back, using the solution and proving that it satisfies the equation? How can I be sure that there are no other solutions? Albmont 18:05, 4 January 2007 (UTC)
Indeed, the SDE $\mathrm{d} X_{t} = \mathrm{d} W_{t}$ has a one-parameter family of solutions, parametrized by the initial condition $x_{0}$. In other words, $X_{t} = x_{0} + W_{t}$ (observing the convention that a Wiener process/Brownian motion starts at the origin). The statement that your proposed, $X_{t} \sim x_{0} + \mathrm{N} (0, t)$ in fact follows from the general formula for the distribution of increments of a Wiener process/Brownian motion: for s < t, $X_{t} - X_{s} \sim \mathrm{N}(0, t - s)$. If you are concerned about whether the Wiener process is the unique process that satisfies its defining conditions, or are wondering about existence and uniqueness theorems for solutions of SDEs, you should consult a good text on basic stochastic processes. From my experience, I would recommend any of these three: [2] [3] [4]. I hope that this helps. Sullivan.t.j 20:07, 4 January 2007 (UTC)

## Merge with Brownian motion?

No, only copy, resum, and cite Wiener process as main article. —Preceding unsigned comment added by 201.52.194.78 (talk) 15:24, 4 February 2008 (UTC)

No. The Wiener process is a very important mathematical construct, independent of any applications to physics (although it is still called Brownian motion). The physical Brownian motion on the other hand is something different, which may or may not be modelled by a Wiener process (I think the integral of the Ornstein-Uhlenbeck process is actually a better model).Roboquant (talk) 03:22, 9 March 2008 (UTC)

## To delete or not to delete?

I believe that Subsection 2.1 "Derivation" is of little interest and should be deleted. However, I hesitate to do it myself. Maybe I find it uninteresting only because I am an expert already? Boris Tsirelson (talk) 08:59, 26 September 2008 (UTC)

I think it should be kept for those less expert, but I have revised the sub-sectioning so that "Derivation" does not appear as a heading. Melcombe (talk) 09:32, 3 June 2009 (UTC)

## Moved comment

From this article I couldn't understand what is that Wiener process and where it appears!!!!!!!!!! Lots of references...and you are finaly lost in the sea of specific information!!!! Could you explain in two sentences what means 'continuous-time'?

The above comment was originally placed at head of Talk by 139.222.112.200 at 19:06, 2 December 2008.

## Why nonsense?

To User:ptrf: I did not understand why you treat the two phrases as nonsense. Maybe they should be more clear. Or maybe they are not needed. Boris Tsirelson (talk) 05:03, 3 June 2009 (UTC)

Well, as far as I can understand, the function w is a trajectory. How can it be differentiable? A few lines above ("quantitative properties") the text says w is not differentiable. It's continuous -- so I had a difficulty to understand why few lines below this "never happens". But, maybe, I'm missing something, as often :) ptrf (talk) 06:57, 3 June 2009 (UTC)
PS. Maybe I could have found a better wording for my comment, sorry if it sounded too harsh.
I see. Well, probably you are right deleting that; the article is not the right place to explain the point, and probably other readers will not understand it, too. But anyway let me explain here (on the talk page) what I had in mind. We may consider the local time (defined as the density of the pushforward measure) for a smooth function too; and the density is discontinuous, unless the given function is monotone. In this sense, the continuity of the local time is another manifestation of non-smoothness of the trajectory. Moreover, it is interesting to see the conflict between good behavior of a function and good behavior of its local time. Boris Tsirelson (talk) 07:26, 3 June 2009 (UTC)
Indeed. Now I think the remark has its place in the article. What about putting what you've just said at the end of the paragraph "local time"? ptrf (talk) 07:42, 3 June 2009 (UTC)
You'd better do it yourself; then probably it will be easier to understand. Boris Tsirelson (talk) 11:12, 3 June 2009 (UTC)
OK, I'll try, but I thought the contrary hold true. ptrf (talk) 11:47, 3 June 2009 (UTC)
Nice. (Thus you refute yourself.) :) Boris Tsirelson (talk) 13:45, 3 June 2009 (UTC)

## Continuity

From the article: "W_t is almost surely continuous"

Is this "Almost surely, W is continuous." or "For all t, W_t is almost surely continuous at t." ? —Preceding unsigned comment added by JumpDiscont (talkcontribs) 21:03, 5 October 2009 (UTC)

It is the former: "Almost surely, W is continuous." Boris Tsirelson (talk) 07:02, 6 October 2009 (UTC)

## running maximum distribution

I added the joint distribution of the maximum and the Wiener process and included an integral to get the unconditional distribution of the maximum. I calculated this by hand, so I would appreciate if someone else confirmed it. I haven't seen it anywhere on the net. 108.35.46.175 (talk) 03:00, 11 April 2012 (UTC)

Oops, no, something must be wrong with it. The (unconditional) distribution of $M_t$ is much easier to get (than the joint distribution) by the reflection principle. It is distributed like $|B_t|.$ Boris Tsirelson (talk) 05:57, 11 April 2012 (UTC)
Yeah I fixed it. Makes sense now. Mathematica to the rescue. — Preceding unsigned comment added by 71.251.222.105 (talk) 13:58, 11 April 2012 (UTC)
OK. Boris Tsirelson (talk) 18:27, 11 April 2012 (UTC)

## Thanks

I just want to thank User:Cyp for his nice new picture. Boris Tsirelson (talk) 17:38, 2 December 2009 (UTC)

Glad you like it. (: Κσυπ Cyp   21:43, 2 December 2009 (UTC)

## Fourier series

I deleted a section on the Fourier series. The formula given looked very wrong, and no reference was given. If someone knows the correct version, please update. 130.235.3.80 (talk) 10:29, 30 October 2013 (UTC)

But see Karhunen–Loève theorem#The Brownian bridge. Boris Tsirelson (talk) 17:15, 30 October 2013 (UTC)

## Running maximum distribution (2)

It should be stated that the unconditional distribution function of the running maximum is defined only on [0,+∞). --Baroc (talk) 22:57, 26 February 2014 (UTC)

OK, I did so. (Density you mean, not distribution function.) But now it looks rather repetitive, since the same restriction is mentioned on the previous display. Boris Tsirelson (talk) 06:41, 27 February 2014 (UTC)