Sufficient conditions for a stochastic process to be continuous?

Question

Let $(X_t)_{t\ge 0}$ be a stochastic process, such that $X_t-X_s \xrightarrow{w} \delta_0$ as $t\to s$. Is that a sufficient condition for $(X_t)_{t\ge 0}$ to be continuous? If not, can you provide what conditions should be needed. Also, can you provide a reference where I can read about continuity of stochastic processes?

Answer 1

It is no different than the continuity of an ordinary function with the exception that you can assign a probability to the event that a given stochastic process is continuous.

Let $X(t,\omega)$ be a stochastic process. The mapping $t \rightarrow X_t(\omega)$ defines a path for each $\omega$. When one talks about the continuity of $X$, he/she means the continuity of these paths. For continuity of a certain path, we require that $t_n \rightarrow t$ imply $X_{t_n}(\omega) \rightarrow X_t(\omega)$.

If $P\{\omega: t\rightarrow X_t(\omega) \text{ is continuous}\} = 1$, then we say $X$ is almost surely continuous.

Answer 2

Consider the poisson point process with constant rate $1$. It is true that $X_t-X_s \xrightarrow[]{w}\delta_0$ but the process is not continuous (It has jumps of size $1$).

To obtain continuity of the stochastic processes you need to prove that the oscilation tends to zero, that is:

$$ \Bbb{P}(\sup_{|t-s|<\delta} |X_t - X_s|> \epsilon)\xrightarrow[\delta \to 0]{} 0$$

A good reference book in this subject is Billingsley https://books.google.com.br/books/about/Convergence_of_Probability_Measures.html?id=GzjbezrsrFcC&redir_esc=y

See the chapter that treat the space $D$ of discontinuous processes.