Consider a stochastic process *X* of the form

for a standard Brownian motion *W* and predictable process , defined with respect to a filtered probability space . For this to make sense, we must assume that is almost surely finite at all times, and I will suppose that is the filtration generated by *W*.

The question is whether the drift can be backed out from knowledge of the process *X* alone. As I will show with an example, this is not possible. In fact, in our example, *X* will itself be a standard Brownian motion, even though the drift is non-trivial (that is, is not almost surely zero). In this case *X* has exactly the same distribution as *W*, so cannot be distinguished from the driftless case with by looking at the distribution of *X* alone.

On the face of it, this seems rather counter-intuitive. By standard semimartingale decomposition, it is known that we can always decompose

for a *unique* continuous local martingale *M* starting from zero, and unique continuous FV process *A*. By uniqueness, and . This allows us to back out the drift and, in particular, if the drift is non-trivial then *X* cannot be a martingale. However, in the semimartingale decomposition, it is required that *M* is a martingale *with respect to the original filtration* . If we do not know the filtration , then it might not be possible to construct decomposition (2) from knowledge of *X* alone. As mentioned above, we will give an example where *X* is a standard Brownian motion which, in particular, means that it is a martingale under its natural filtration. By the semimartingale decomposition result, it is not possible for *X* to be an -martingale. A consequence of this is that the natural filtration of *X* must be strictly smaller than the natural filtration of *W*.

The inspiration for this post was a comment by Gabe posing the following question: If we take to be the filtration generated by a standard Brownian motion *W* in , and we define , can we find an -adapted such that the filtration generated by is smaller than ? Our example gives an affirmative answer. (more…)