Question on notation of stochastic processes

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  • #1
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Hello,

when we have a deterministic signal f:ℝ→ℝ that is square integrable we can typically write [itex]f \in L^2(\mathbb{R})[/itex].

However, what if [itex]\{ f(t): \; t\in \mathbb{R} \}[/itex] are random variables, i.e. f is a continuous-time stochastic process?

What is the notation to denote the space of "square integrable" stochastic processes?
Here for square integrable I mean the following:

[tex]E\left\{ \int_{-\infty}^{+\infty} |f(t)|^2 dt \right\} < \infty[/tex]

where E denotes the expected value.
 

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  • #2
My book, Stochastic Integration by Kuo referes to a stochastic process most explicitly as the space is [itex] L_{ad}^2([a,b] \times \Omega)[/itex] as the space all stochastic processes [itex]f(t,\omega), a \leq t \leq b, \omega \in \Omega[/itex] live in such that

1) [itex]f(t,\omega)[/itex] is adapted to filtration [itex]\lbrace\mathscr{F}_t \rbrace[/itex]

2)[itex]\int\limits_a^b{E|f(t,\omega)|^2dt} < \infty[/itex]

I've seen this before in other books... the space may also written on the entire positive real line. i.e. [itex] L^2(\mathbb{R}_+ \times \Omega)[/itex]
 
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Hello.
Thanks for your help. Could you please explain what does the subscript "ad" in [itex]L_{ad}^2[/itex] mean?

I don't quite understand point 1) either, because I don't know what is a filtration in this context.
 
  • #4
I am currently away and don't have the book with me. I was wondering if it may be a typo, since they use a and b everywhere else? I will respond when I can look at the book.

I presume the filtration is referring to the natural filtration [itex]\mathscr{F}_t := \sigma{ \lbrace f(s,\omega); s \leq t \rbrace }[/itex] so that may help, but I am not 100% sure and do not want to mislead you. I will make sure when I get a chance to look at the book again.
 
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  • #5
So it is not directly referring to the natural filtration after all,

It was referring to any filtration [itex]\mathscr{F}_t[/itex] that satisfied

1) [itex] \forall t, B(t)[/itex] is a [itex]\mathscr{F}_t-[/itex]measurable.

2) [itex] \forall s \leq t[/itex], the random variable [itex]B(t) - B(s)[/itex] is independent of the [itex]\sigma-[/itex]field [itex]\mathscr{F}_s[/itex]

as for the "ad" subscript, the answer is much much less clear and the answer appears to be buried in a multiple-page proof based on Ito's original paper on the stochastic integrals. It does not appear to be a typo as it is listed in the notation for the book, but the label says nothing beyond "a class of integrands".
 

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