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)Reply

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)Reply

I think that we had better put

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

covariance

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

and its derivative.

${\displaystyle a=t0<t1<...<tn=b}$

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

${\displaystyle (I)\int _{a}^{b}W(t)\,dW(t)=(W^{2}(b)-W^{2}(a))/2-(b-a)/2.}$

${\displaystyle (S)\int _{a}^{b}W(t)\,dW(t)=(W^{2}(b)-W^{2}(a))/2.}$

—Preceding unsigned comment added by Jackzhp (talk • contribs)

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 (talk • contribs)

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)Reply

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

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)Reply

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)Reply

Indeed, the SDE ${\displaystyle \mathrm {d} X_{t}=\mathrm {d} W_{t}}$ has a one-parameter family of solutions, parametrized by the initial condition ${\displaystyle x_{0}}$. In other words, ${\displaystyle X_{t}=x_{0}+W_{t}}$ (observing the convention that a Wiener process/Brownian motion starts at the origin). The statement that your proposed, ${\displaystyle X_{t}\sim x_{0}+\mathrm {N} (0,t)}$ in fact follows from the general formula for the distribution of increments of a Wiener process/Brownian motion: for s < t, ${\displaystyle X_{t}-X_{s}\sim \mathrm {N} (0,t-s)}$. 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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

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.

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)Reply

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)Reply

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)Reply

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)Reply

You'd better do it yourself; then probably it will be easier to understand. Boris Tsirelson (talk) 11:12, 3 June 2009 (UTC)Reply

OK, I'll try, but I thought the contrary hold true. ptrf (talk) 11:47, 3 June 2009 (UTC)Reply

Nice. (Thus you refute yourself.) :) Boris Tsirelson (talk) 13:45, 3 June 2009 (UTC)Reply

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 (talk • contribs) 21:03, 5 October 2009 (UTC)Reply

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

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)Reply

Oops, no, something must be wrong with it. The (unconditional) distribution of ${\displaystyle M_{t}}$ is much easier to get (than the joint distribution) by the reflection principle. It is distributed like ${\displaystyle |B_{t}|.}$ Boris Tsirelson (talk) 05:57, 11 April 2012 (UTC)Reply

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)Reply

OK. Boris Tsirelson (talk) 18:27, 11 April 2012 (UTC)Reply

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

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

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)Reply

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

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)Reply

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)Reply

It is written that W has Gaussian increments: W_t+u - W_t is normally distributed with mean 0 and variance u, W_t+u−W_t ~ N(0, u)

However, later it is written that variance is t. Perhaps then notation should be W_t+u−W_t? ~ N(0, sqrt(u)) (assuming usual normal distribution notation) --92.42.31.61 (talk) 16:02, 2 January 2016 (UTC)Reply

No, the "usual normal distribution notation" is N(μ, σ²), not N(μ, σ) (as you probably believe); see Normal_distribution#Notation. Boris Tsirelson (talk) 18:08, 2 January 2016 (UTC)Reply

I am a learner, I myself need a reference for this sentence in Paragraph 3: "It also forms the basis for the rigorous path integral formulation of quantum mechanics " Can anyone help? — Preceding unsigned comment added by Davy2016 (talk • contribs) 03:42, 23 December 2016 (UTC)Reply

In property 3, shouldn't the mean be 0? (That's what the article said before 2017)

In the section "Qualitative properties", it says that for any ${\displaystyle \epsilon >0}$, ${\displaystyle w(t)}$ is almost surely not ${\displaystyle ({\tfrac {1}{2}}+\epsilon )}$-Hölder continuous. But it seems that we can do better: perhaps ${\displaystyle w(t)}$ is almost surely not ${\displaystyle {\tfrac {1}{2}}}$-Hölder continuous? I have seen this claim in a comment to this Math Overflow question without proof. Is this result correct? 129.104.241.198 (talk) 15:49, 3 September 2023 (UTC)Reply

@129.104.241.198 There is actually a subtle difference between the case ${\displaystyle \alpha =1/2+\varepsilon }$ and ${\displaystyle \alpha =1/2}$: While it is true for all ${\displaystyle \alpha \geq 1/2}$ that ${\displaystyle W}$ is almost surely not (locally) ${\displaystyle \alpha }$-Hölder continuous at a fixed time ${\displaystyle t}$, it only holds for ${\displaystyle \alpha >1/2}$ that ${\displaystyle W}$ is nowhere ${\displaystyle \alpha }$-Hölder continuous. In fact, in the case ${\displaystyle \alpha =1/2}$, the latter is not true, that is, there is a non-zero probability that a path of Brownian motion is ${\displaystyle 1/2}$-Hölder continuous somewhere. But the point ${\displaystyle t}$ where this happens is different from path to path.

I changed the corresponding statements in the article to make this more clear. Simonbiene (talk) 13:42, 31 July 2026 (UTC)Reply

I find both the title and the structure of the first section "Characterisations of the Wiener process" unsatisfying. The title is too long and the section is currently a collection of paragraphs discussing all sorts of characterisations in prose. Moreover, some topics occur twice in the article, such as Donsker's theorem (section 1 and 2) and Karhunen–Loève (section 1 and "Wiener representation").

I would find it more intuitive to call it simply "Definitions" and present the different characterisations in separate subsections as precise statements. I would also suggest removing the duplicate sections and include the information from there in the corresponding subsection of "Definitions".

I volonteer to make these changes, if no one disagrees. Simonbiene (talk) 21:30, 30 July 2026 (UTC)Reply