Stochastic process
Definition: Stochastic process A stochastic process is a random varible $X(t)$ depending on a parameter $t\in T$ For every fixed $t$ $X$ is a map from the sample space $\Omega$ to $\Reals$ (a random variable) so that $$X: \Omega\times T \to \Reals$$
Definition: Trajectory If we keep $\omega$ fixed the mapping
$$t \to X(\omega,t)$$ is called a trajectory or sample path
A multivariate stochastic process is defined analogous as $X(t) = (X_1(t), …, X_d(t))$ and $$X: \Omega\times T \to \Reals^d.$$
Usually one characterizes a stochastic process by looking at correlation of the random variable at different points in time.
Definition: Joint probability distribution A joint probability distribution of order $m$ is defined as $$P(B_1, t_1; …; B_m, t_m)=\mu(X(t_1)\in B_1, …, X(t_m)\in B_m).$$ It gives the probability that $X$ takes on some value in $B_1$ at $t_1$ and so on…
Definition: Family of finite joint probability distributions The family of finite joint probability distributions is the set of all joint probability distributions for all orders $m$ and at all discrete times $t$ and all Borel sets $B$
A joint probability distribution satisfies the Kolmogorov consistency conditions: $$P(\Reals^d,t)=1$$ $$P(B_1,t_1;…)\geq 1$$ $$P(B_1,t_1; …;B_{m-1},t_{m-1};\Reals^d,t_m)=P(B_1,t_1; …;B_{m-1},t_{m-1})$$ In addition the joint probability distribution is invariant under permutations of the arguments
Theorem of Kolmogorov: If a family functions fulfills these conditions, there exists a probability space and a stochastic process $X(t)$ such that the joint probability distributions on $X$ build the given family of functions. However the process $X(t)$ is not unique