# Autocovariance of increments of a semimartingale

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 $$dX_t dX_s$$?

## 1 Answer

**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.