Question on notation of stochastic processes

In summary, the space of "square integrable" stochastic processes is written as L_{ad}^2([a,b] \times \Omega), where [a,b] is the domain and \Omega is the range. The notation for the space may also be written on the entire positive real line.
  • #1
mnb96
715
5
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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  • #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.
 
  • #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".
 

1. What is the purpose of notation in stochastic processes?

The purpose of notation in stochastic processes is to provide a standardized and concise way of representing and communicating mathematical concepts and formulas related to these processes. It allows for easier understanding and communication among scientists and researchers, as well as facilitating the development and analysis of new models and theories.

2. What are the common notations used in stochastic processes?

Some of the common notations used in stochastic processes include random variables (represented by uppercase letters such as X or Y), probability distributions (represented by lowercase letters such as p or q), and time (usually denoted by lowercase t). Other commonly used symbols include subscripts, superscripts, and mathematical operators such as summation and integration.

3. How do notations differ between discrete and continuous stochastic processes?

In discrete stochastic processes, notations typically represent discrete values such as integers or binary outcomes. In contrast, notations in continuous stochastic processes often involve continuous variables such as real numbers or time. Additionally, the mathematical operators used in notations may also differ between discrete and continuous processes.

4. Can notations in stochastic processes be simplified?

Yes, notations in stochastic processes can be simplified by using shorthand notations or abbreviations. For example, the notation E[X] is often used to represent the expected value of a random variable X, and Var(X) is used to represent the variance of X. However, it is important to ensure that the simplified notation is still clear and unambiguous.

5. Is there a universal notation for stochastic processes?

No, there is no universal notation for stochastic processes. Different fields and researchers may use different notations, and even within the same field, there may be variations in notation depending on the specific context or application. It is important to be familiar with the common notations used in your specific area of research.

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