$M$ be a Gaussian martingale with continuous sample paths, such that $M_0=0$. I want to show that, for every $t \geq 0$ and every $s >0$, the random variable $M_{t+s}-M_t$ is independent of $\sigma(M_r, 0\leq r \leq t)$.

I guess I need to show $E[M_r (M_{t+s}-M_t)]=E[M_r] E[(M_{t+s}-M_t)]$.

I appreciate any hints...

  • $\begingroup$ $E[(M_{t+s}-M_t)M_r]=E[M_rE[M_{t+s}-M_t|\sigma(M_r,0\le r\le t)]]=0$. Hence $M_{t+s}-M_t$ and $(M_r,0\le r\le t)$ are uncorrelated and independent. $\endgroup$ – JGWang May 6 '17 at 3:32
  • $\begingroup$ @JGWang I forgot $E[M_r]=0$. Thanks! $\endgroup$ – Siskaa May 7 '17 at 13:43

For $0=r_0<r_1<r_2<\cdots<r_n\le t$, the random vector $(M_{r_1}-M_{r_0},M_{r_2}-M_{r_1},\ldots,M_{r_n}-M_{r-{n-1}},M_{t+s}-M_t)$ has the multivariate normal distribution with zero mean vector and diagonal covariance matrix. It follows that $(M_{r_1}-M_{r_0},M_{r_2}-M_{r_1},\ldots,M_{r_n}-M_{r-{n-1}})$ is independent of $M_{t+s}-M_t$. Because $\sigma(M_r,0\le r\le t)$ is generated by events of the form $\cap_{k=1}^n\{M_{r_k}-M_{r_{k-1}}\in B_k\}$, for various choices of the $r_k$ and Borel sets $B_k$, the stated independence follows from the monotone class theorem.

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