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. Continue reading “A Process With Hidden Drift”