Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question on notation of stochastic processes

  1. Oct 13, 2013 #1

    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
    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".
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook