Almost Sure

20 April 20

A Process With Hidden Drift

Consider a stochastic process X of the form

\displaystyle  X_t=W_t+\int_0^t\xi_sds, (1)

for a standard Brownian motion W and predictable process {\xi}, defined with respect to a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in{\mathbb R}_+},{\mathbb P})}. For this to make sense, we must assume that {\int_0^t\lvert\xi_s\rvert ds} is almost surely finite at all times, and I will suppose that {\mathcal F_\cdot} is the filtration generated by W.

The question is whether the drift {\xi} 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 {\xi} is non-trivial (that is, {\int\xi dt} 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 {\xi=0} 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

\displaystyle  X=M+A (2)

for a unique continuous local martingale M starting from zero, and unique continuous FV process A. By uniqueness, {M=W} and {A=\int\xi dt}. This allows us to back out the drift {\xi} 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 {\mathcal F_\cdot}. If we do not know the filtration {\mathcal F_\cdot}, 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 {\mathcal F_\cdot}-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 {\mathbb F} to be the filtration generated by a standard Brownian motion W in {(\Omega,\mathcal F,{\mathbb P})}, and we define {\tilde W_t=W_t+\int_0^t\Theta_udu}, can we find an {\mathbb F}-adapted {\Theta} such that the filtration generated by {\tilde W} is smaller than {\mathbb F}? Our example gives an affirmative answer. (more…)

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