Let $(\Omega,\mathcal{F})$ be a measurable space, carrying a stochastic process $X=X_{t≥0}$ with state space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $\mathcal{F}_t = \sigma(X_s:s≤t)$. Assume that trajectories $t\mapsto X_t(w)$ are continuous for all $w\in\Omega$. Prove or disprove : $(\mathcal{F}_t)$ is right continuous.
If a stochastic process is path-wise continuous then its filtration is right-continuous?
1
$\begingroup$
probability-theory
stochastic-processes