Example of natural filtration which is not right-continuous

Question

Can anyone please give me an example of a process $X$ whose natural filtration satisfies $(\mathscr{F}_{t}^{X_{+}})_{t \ge 0} \neq (\mathscr{F}_{t}^{X})_{t \ge0}$?

Answer 1

Let $(W_n)_{n \in \mathbb{Z}_+}$ be a simple symmetric random walk on $\mathbb{Z}_+$ starting at $0$. Interpolate $W$ linearly on each time interval $[n,n+1]$. This defines a continuous and piecewise affine process $(W_t)_{t \ge 0}$. Its natural filtration satisfies $\mathcal{F}^W_t = \mathcal{F}^W_{\lceil t \rceil}$ for each $t \ge 0$. It is right-discontinuous at each integer time $n$. Indeed, the value of the increment $W_{n+1}-W_n$ is unknown at time $n$ (since $W_{n+1}-W_n$ is independent of $\mathcal{F}^W_n$) but is known immediatly after (since for every $\epsilon \in (0,1)$, $W_{n+1}-W_n = (W_{n+\epsilon}-W_n)/\epsilon$ is $\mathcal{F}^W_{n+\epsilon}$-measurable).