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Question on notation of stochastic processes

  1. Oct 13, 2013 #1
    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.
     
  2. jcsd
  3. Oct 19, 2013 #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]
     
    Last edited: Oct 19, 2013
  4. Oct 19, 2013 #3
    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.
     
  5. Oct 21, 2013 #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.
     
    Last edited: Oct 21, 2013
  6. Oct 22, 2013 #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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