Why Girsanov's theorem used here?

Question

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.

Answer 1

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).

Girsanov theorem is about change of probability measures as you correctly mention too. To me, the author simply means that this implies there exist means of moving from P to a bunch of other equivalent measures (notably one under which under which drift will be zero).

So I guess there are two things:

• Given that P exists, do other equivalent probability measures exist, what are they and how do we move from P to these measures? This is given by Girsanov theorem. Amongst these equivalent measures there exists one under which the drift is zero.

• If the drift of a continuous paths processis zero under a certain measure, then the process will be a martingale under that measure.