Question on notation of stochastic processes

[mnb96]

Hello,

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

However, what if \{ f(t): \; t\in \mathbb{R} \} 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:

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

where E denotes the expected value.

 

[X89codered89X]

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

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

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

I've seen this before in other books... the space may also written on the entire positive real line. i.e. L^2(\mathbb{R}_+ \times \Omega)

 

[mnb96]

Hello.
Thanks for your help. Could you please explain what does the subscript "ad" in L_{ad}^2 mean?

I don't quite understand point 1) either, because I don't know what is a filtration in this context.

 

[X89codered89X]

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 \mathscr{F}_t := \sigma{ \lbrace f(s,\omega); s \leq t \rbrace } 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.

 

[X89codered89X]

So it is not directly referring to the natural filtration after all,

It was referring to any filtration \mathscr{F}_t that satisfied

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

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

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".