0
$\begingroup$

I am studying a stochastic process $(X_t)$ in continuous time which has independent increments. For $\mathscr{F}_t$ being the natural filtration I would like if $$ \mathop{\mathbb{E}}[X_t-X_s \vert \mathscr{F}_s] = \mathop{\mathbb{E}}[X_t-X_s] $$ whenever $s \leq t$. Is it true and do we in general have that $X_t-X_s$ is independent of $\mathscr{F}_r$ for $r \leq s \leq t$?

$\endgroup$
1
$\begingroup$

Yes, it is true. You can prove this by showing that $(X_t-X_s)^{-1}(A)$ is independent of $X_{r_1}^{-1}(A_1) \cap X_{r_2}^{-1}(A_2) ... \cap X_{r_N}^{-1}(A_N)$ whenever $N$ is a positive integer, $r_1<r_2<\cdots<r_N\leq r$ and each $A_i$ is a Borel set.

$\endgroup$
  • $\begingroup$ Thank you. What you are hinting at here is it to use that $\mathscr{F}_r$ is generated by intersections on the form $X_{r_1}^{-1}(A_1) \cap ... \cap X_{r_N}^{-1}(A_N)$ as you write and the use that it is enough to show independence of a generator of the sigma algebra? If so, can you be a bit more explicit on how to continue the proof from here? It will be appreciated a lot. $\endgroup$ – J. Goles Sep 3 '19 at 6:45
  • $\begingroup$ @J.Goles One way is to use the $\pi -\lambda$ theorem. Consider $\{ F: \in \mathcal F_r: P( E\cap F)=P(E)P(F)$ where $E=(X_t-X_s)^{-1}(A)$. This is a $\lambda system and we have proved that it contains a$\pi$ system which generates $\mathcal F_r$. $\endgroup$ – Kavi Rama Murthy Sep 3 '19 at 7:34
  • $\begingroup$ Thank you. I will look further in to that. $\endgroup$ – J. Goles Sep 9 '19 at 6:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.