How Girsanov's Theorem Connects Measure Changes to Martingale Drifts
Summary
The document clarifies why Girsanov's theorem may be cited when discussing price processes with zero drift under a martingale measure. Girsanov describes how changing between equivalent probability measures alters a process's drift; in an appropriate measure, the drift can become zero. A separate martingale result then relates zero drift to martingale behavior for continuous-path processes.
The answer cautions that the simple zero-drift criterion does not apply in the same way to jump processes, whose martingale representations can include compensator terms. It distinguishes the role of choosing or transforming to a measure from the role of characterizing a martingale once that measure is fixed. The explanation is conceptual and does not derive the theorem or state the conditions required for a particular measure change, so formal applications need additional assumptions.
Key ideas
- Girsanov's theorem describes how drift changes under an equivalent change of probability measure.
- A suitable measure can make a price process have zero drift.
- For continuous-path processes, zero drift is associated with martingale behavior under the relevant measure.
- Jump processes can have compensator terms even when they are martingales.
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# Why Girsanov's theorem used here?
# 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.
## Answer by Quantuple (score 1)
https://quant.stackexchange.com/a/40854
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.