2
$\begingroup$

Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.

I'm reading a paper [1], which uses a notation for stochastic processes that doesn't seem to be standard.

Now we generalize the squared covariance and define the square of conditional covariance, given two real-valued stochastic processes $U(·)$ and $V(·)$. We obtain an interesting result when $U$ and $V$ are independent Weiner processes. First, to center the random variable $X$ in the conditional covariance, we need the following definition. Let $X$ be a real-valued random variable and ${U (t):t ∈ ℝ}$ a real-valued stochastic process, independent of $X$. The $U$-centered version of $X$ is defined by

$$X_U = U (X) − \int^∞_{-∞} U (t) dFX(t) = U (X) − E[U (X)|U]$$

In my experience a stochastic process is a collection of random variables $\{X_t\}$... What is the intuition behind a stochastic process $U(·)$? Also, they have both $U(X)$ and $U(t)$ which I can't make heads or tails of.

Cheers

$\endgroup$
1
$\begingroup$

$U(X)$ just means the stochastic process at a random time $X$. So you have two different random things, the stochastic process (collection of random variables) $\{U(t)\}$, and the random time $X$ picks out a particular one $U(X)$.

For instance, imagine you pick a random book from the shelf and open it on a random page. Then $\{U(t)\}$ is the book, $X$ is the page number, and $U(X)$ is the text on that page.

$\endgroup$
  • $\begingroup$ So $X_U$ is the text on the page, normalized against the average text from all books on that same page. Thanks for the example. :) $\endgroup$ – Scott Dec 8 '16 at 8:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.