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.

• Given that Q is a martingale measure for a certain continuous paths process, then this process will exhibit no drift under Q by the martingale representation theorem.

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.

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

Girsanov theorem is about change of probability measures as rightly you 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. Combined to the previous representation theorem this make the latter measure Q, the martingale measure we were looking for.

So you are right, Girsanov only is useful if you want to relate P to Q. The fact that there is no drift under Q is indeed related to the martingale representation theorem of continuous paths martingales.

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.

• Given that Q is a martingale measure for a certain continuous paths process, then this process will exhibit no drift under Q by the martingale representation theorem.