# Why Girsanov's theorem used here?

It is written in Bjork's ArbitrageTheoryInContinuousTime that

... Assume a martingale measure Q exists. This implies (see the Girsanov theorem) that the price processes have zero drift under $Q$ ...

It is written in the third edition on page 141 at the bottom.

I don't understand what he's talking about. It is a known, general result that stochastic differentials are martingales if and only if they have no $\text{dt}$-term. It's got nothing to do with Girsanov's theorem. Girsanov's theorem is about how when we change from $P$ to $Q$, we preserve $\sigma$ but the drift-term changes to something else.

So why is he referring to the Girsanov theorem here, rather than the general result of "no dt-term $\iff$ martingale". In fact, Girsanov seems completely irrelevant here.

The first result you are alluding to is known as the martingale representation theorem. More specifically, what you say holds for continuous paths processes. For jump processes, there can and will a $dt$ term in their martingale representation (compensator).