# Stochastic Process Notation

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

$U(X)$ 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.
• 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. :) – Scott Dec 8 '16 at 8:41