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)] = σ2min(s,t)

covariance

γ(s,t) = E[(X(s) − m(s))(X(t) − m(t))] = ...

and its derivative.

a = t0 < t1 < ... < tn = b

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 (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)

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 dXt = dWt has a one-parameter family of solutions, parametrized by the initial condition x0. In other words, Xt = x0 + Wt (observing the convention that a Wiener process/Brownian motion starts at the origin). The statement that your proposed, Xt˜x0 + N(0,t) in fact follows from the general formula for the distribution of increments of a Wiener process/Brownian motion: for s < t, Xt − Xs˜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)

[edit] 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)