Say that $X_t$ is an Itō process with \begin{equation} dX_t = \mu_t dt + \sigma_t dW_t \end{equation} where $\mu_t$ and $\sigma_t$ are adapted processes.

Is it always true that \begin{equation} E[dX_t dX_s] = 0, \quad t\neq s \end{equation} i.e. the increments of $X_t$ are always independent? At first glance I would say yes since $dX_t dX_s \propto dtds$ which will integrate to zero, but is there something more that can be said about $dX_t dX_s$?


**please correct me if the math is wrong!!

I think upon breaking down the products $E(dX_tdX_s)$, we have the $dtds$, $dtdW_s$ terms which all turns out to be 0. It leaves $E(dW_tdW_s)$ which comes down to sharing the same wiener process $dW = \sqrt{dt}Z$, where Z follows N(0,1).

However note that since we're calculating expecation, $dW_t$ and $dW_s$ is basically the same thing under integration, because they are both denoting a change in dt. so $E(dW_tdW_s)$ is the same as $E(dW_tdW_t)$, which equals 0.

Hence it seems like the equation always equals to 0.


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