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Does there exist a stochastic process $X_t$, $t \in [0,\infty)$, such that $X_t$ is distributed according to some distribution $f(x)$ that possesses finite variance and such that $X_t$ and $X_s$ are independent for all $s \neq t$?

Unlike other questions similar to this one I do not demand continuity in sampling paths.

If it exists what would a measure of the sampling paths look like? What would the paths look like?

My intuition is that such a processes probably doesn't exist based on my understanding of white noise. I.e. white noise in continuous time would have the above properties, but my understanding is that we must integrate over white noise to have something that makes sense and why we must write SDEs as

$$dx = \mu dt +\sigma dB$$

rather than

$$\dot{x} = \mu + \sigma \phi$$

where $\phi\sim dB/dt$ is the white noise as $\phi$ is not really a coherent concept.

Can anyone help me out here?

Rodrigo de Azevedo
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user3353819
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  • @RodrigodeAzevedo I mention white noise in my post. My understanding is that formally it is not a solid concept and hence why we use Brownian motions in SDEs, not white noise – user3353819 Jun 09 '19 at 14:12
  • Communications theory is built on white noise and a space probe beyond Pluto can transmit at 1 Watt and people on Earth can decode the message. If engineers waited for mathematicians to make things rock solid, progress would slow down. – Rodrigo de Azevedo Jun 09 '19 at 14:15
  • @RodrigodeAzevedo I mean, fair enough, but I'm asking if it formally exists as a stochastic process, i.e. if there is an actual existing function $X_t$ with these properties? Sure, in the real world where we use finite deltas the concept is fine. And if I'm wrong about anything, that's fine, but it does need to be argued. – user3353819 Jun 09 '19 at 14:19
  • If white noise is not admissible, then I can't think of anything. – Rodrigo de Azevedo Jun 09 '19 at 14:20
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    @RodrigodeAzevedo: White noise is not really a stochastic process, but rather a generalized stochastic process. Its mathematical construction is not on $\mathbb R^{[0,\infty )}$ (that is what we would expect for a stochastic process), but on the space of distribution. But indeed, it would have been a good candidate :-) – Surb Jun 09 '19 at 14:30
  • @user3353819: Such a process doesn't exist (I don't know the argument, but it's mention in the book Stochastic Differential Equations of B. Oksendal at the beginning of the chapter on Itô integral (if you have the 6th edition, it's chapter 3). – Surb Jun 09 '19 at 14:34
  • @Surb Ok, good. Thanks for the reference - I'll look it up. – user3353819 Jun 09 '19 at 14:36

1 Answers1

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Yes, of course, as guaranteed by the Kolmogorov extension theorem. For instance, one can take $\lambda^{\mathbb R}$ where $\lambda$ is Lebesgue measure on $[0,1]$. But, as the Wikipedia article makes clear, it's not very useful

Kolmogorov's extension theorem applies when $T$ is uncountable, but the price to pay for this level of generality is that the measure $\nu$ is only defined on the product $\sigma$-algebra of $(\mathbb {R} ^{n})^{T}$, which is not very rich.

(Here $T$ would be your $[0,\infty)$.) This subject is immensely technical, and is discussed fully in chapter 7 of Bogachev's Measure Theory. I hesitate to summarize the crux of what goes wrong, but one thing is that your underlying sample space (something like $\mathbb R^{\mathbb R})$ is not separable; this meshes poorly with the countable-ness built into $\sigma$-algebras.

kimchi lover
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  • Thanks, but I am now somewhat confused. I have conflicting responses. I'm sure the answer is deeply technical. But my immediate questions is then what does Oksendal mean when he describes white noise as not existing as a "reasonable" stochastic process? If you are correct, in what sense is @Surb incorrect? – user3353819 Jun 10 '19 at 13:05
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    The entire world is confused by this topic, so you are definitely not alone. There are 2 ways to think about stochastic processes: a method of randomly picking an object like a function of time, or a method of putting a joint distribution together from marginals. In the case of white noise, one problem is in deciding what space the sample paths are to live in and how to define a probability structure on that space. The other route, the Kolmogorov extension route is superficially easy, as I tried to show in my answer, but actually useless because of $\sigma$ field problems. – kimchi lover Jun 10 '19 at 13:29