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 $r \leq s \leq t$?


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.

  • $\begingroup$ 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

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