Consider a non increasing, caglad process $(X_t)_{t\geq0}$ such that, for each $t$, the distribution function $F_t(x):=P(X_t\leq x)$ is a continuous function of (real) $x$. Are there any sufficient conditions for $(X_t)_{t\geq0}$ to be continuous in $t$?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
I don't know if continuity of the density functions gives you anything. For example, think about a pure jump process with smooth density function of the jump distribution.
You may be familiar with Kolmogorov's lemma, to be found in many books on stochastic processes. This is the first thing to try. It states that if $\mathbb E |X_s - X_t|^{\alpha} \leq C |s-t|^{n+\varepsilon}$, (with $n = 1$ in this case) for some constants $\alpha, C, \varepsilon$, and all $s, t \geq 0$, then there exists a continuous version of $X$ that is in fact Hölder continuous of order $\theta$ for any $\theta < \varepsilon / \alpha$. See [Rogers, Williams 1994, Volume I], or many other references, like http://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem.