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".