The sample paths of a stochastic process.

Question

I´m trying to understand the notion of a stochastic process so I came with the following problem. Let $X$ be a standard normally distributed random variable and form the stochastic process $$Y_{t}=X+2t.$$

For this particular example, I have two questions:

1.- How can I describe the simple paths of the process?

2.- What is the probability that $Y_{t}=0$ for some $t\in \mathbb{N}$?

I´ve never came across with particular examples of stochastic processes, so I hope some illumination with this definitions.

Thank you.

Answer 1

When talking about stochastic processes you should always specify the time (or "index") set: $\mathbb R,$ $\mathbb R^+$, $\mathbb Z,$ $\mathbb N,$ or what?

Assuming you mean $\mathbb N$, your sample paths are arithmetic sequences increasing 2 at a time, starting at $Y_1 = 2+X$ where $X\sim N(0,1)$. For each $t\in\mathbb N$, $P(Y_t=0) = 0$, and by countable additivity, $P(\exists t\in\mathbb N : Y_t=0) = 0$. But if your time set is $\mathbb R$, of course for $t=-X/2$ you will have $Y_t=0$.

One way to think about stochastic processes is that they are collections of random variables, indexed by $t$; another is that are random functions of $t$. The latter is, I think, easier for beginners to think about, and you should develop some intuition and comfort level working with both points of view.