Ways for characterizing stochastic processes

I was wondering what are some general approaches to characterize a stochastic process? Sorry for the vagueness that my questions may have, but let me try to make clear as much as I could that question I have been having for a long time.

1. First of all, I want to understand how detailed a description of a stochatic process needs to be so that it can be said to be a characterization? The law on the trajectory space, or the mapping from the probability space to the trajectory space, or ...? It is similar to the case that a random variable cannot be identified with its law.
2. It is often mentioned that there are two points-of-view that we are taking of a stochastic process, namely (i) a collection of random variables, and (ii) a random function of the index (called a trajectory).

In the second approach, I was wondering what is the general approach to characterize a stochastic process based on the trajectories? We can study the law of a stochastic process on its trajectory space. But the process may have some properties on the sample paths, such as continuity, which cannot be captured by its law.

In the first approach, I am not sure what it really is. The examples that I have seen is just to study the 1-dimensional distributions, such as their moments. Or maybe all the finite dimensional distributions are studied in this approach? Also they don't seem able to fully characterize a stochastic process.

Thanks and regards!

-
1. Finite dimensional distribution do fully characterize a stochastic processes (thanks to KET). $\phantom{x}\\$ 2. But the process may have some properties on the sample paths, which cannot be captured by its law. For example? – Ilya Jan 17 '13 at 17:18
Thanks, @Ilya! Can continuity of its trajectories be captured by its law? – Tim Jan 17 '13 at 17:20
Depends on your definition of the law (which in particular includes the definition of a sample space). – Ilya Jan 17 '13 at 17:24
@Ilya: Do you mean the same as restricting the trajectory space to its subset with the required properties? how detailed a description of a stochatic process needs to be so that it can be said to be a characterization? The law on the trajectory space, or the mapping from the probability space to the trajectory space, or ...? It is similar to the case that a random variable cannot be identified with its law. – Tim Jan 17 '13 at 17:26
What do you precisely mean by saying that a random variable cannot be identified with its law? – Ilya Jan 17 '13 at 17:28