filtration – Almost Sure

In stochastic calculus it is common to work with processes adapted to a filtered probability space ${ (\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}$ . As with probability space extensions, It can sometimes be necessary to enlarge the underlying space to introduce additional events and processes. For example, many diffusions and local martingales can be expressed as an integral with respect to Brownian motion but, sometimes, it may be necessary to enlarge the space to make sure that it includes a Brownian motion to work with. Also, in the theory of stochastic differential equations, finding solutions can sometimes require enlarging the space.

Extending a probability space is a relatively straightforward concept, which I covered in an earlier post. Extending a filtered probability space is the same, except that it also involves enlarging the filtration ${\{\mathcal F_t\}_{t\ge0}}$ . It is important to do this in a way which does not destroy properties of existing processes, such as their distributions conditional on the filtration at each time.

Let’s consider a filtered probability space ${(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}$ . An enlargement

$\displaystyle \pi\colon (\Omega',\mathcal F',\{\mathcal F'_t\}_{t\ge0},{\mathbb P}')\rightarrow(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})$

is, firstly, an extension of the probability spaces. It is a map from Ω′ to Ω measurable with respect to ${\mathcal F'}$ and ${\mathcal F}$ , and preserving probabilities. So ℙ′(π^-1E) = π(E) for all ${ E\in\mathcal F}$ . In addition, it is required to be ${\mathcal F'_t/\mathcal F_t}$ measurable for each time t ≥ 0, meaning that ${\pi^{-1}(E)\in\mathcal F'_t}$ for all ${ E\in\mathcal F_t}$ . Consequently, any adapted process X_t lifts to an adapted process X^∗_t = π^∗X_t on the larger space, defined by X^∗_t(ω) = X_t(π(ω)).

As with extensions of probability spaces, this can be considered in two steps. First, we extend to the filtered probability space on Ω′ with induced sigma-algebra ${\pi^*\mathcal F}$ consisting of sets π^-1E for ${ E\in\mathcal F}$ , and to the filtration ${\pi^*\mathcal F_t}$ . This is essentially a no-op, since events and random variables on the original filtered probability space are in one-to-one correspondence with those on the enlarged space, up to zero probability events. Next, the sigma-algebras are enlarged to ${\mathcal F'\supseteq\pi^*\mathcal F}$ and ${\mathcal F'_t\supseteq\pi^*\mathcal F_t}$ . This is where new random events are added to the event space and filtration.

Such arbitrary extensions are too general for many uses in stochastic calculus where we merely want to add in some additional source of randomness. Consider, for example, a standard Brownian motion B defined on the original space so that, for any times s < t, B_t – B_s is normal and independent of ${\mathcal F_s}$ . Does it necessarily lift to a Brownian motion on the enlarged space? The answer to this is no! It need not be the case that B^∗_t – B^∗_s is independent of ${\mathcal F'_s}$ . For an extreme case, consider the situation where ${(\Omega',\mathcal F',{\mathbb P}')=(\Omega,\mathcal F,{\mathbb P})}$ and π is the identity, so there is no enlargement of the sample space. If the filtration is is extended to the maximum, ${\mathcal F'_t=\mathcal F}$ , consider what happens to our Brownian motion. The increment B_t – B_s is ${\mathcal F'_s}$ -measurable, so is not independent of it. In fact, conditioned on ${\mathcal F'_0}$ , the entire path of B is deterministic. It is definitely not a Brownian motion with respect to this new filtration. Similarly, martingales, submartingales and supermartingales will not remain as such if we pass to this enlarged filtration.

The idea is that, if ${ Y={\mathbb E}[X\vert\mathcal F_t]}$ for random variables X, Y defined on our original probability space, then this relation should continue to hold in the extension. It is required that ${ Y^*={\mathbb E}[X^*\vert\mathcal F'_t]}$ . This is exactly relative independence of ${\mathcal F'_t}$ and ${\pi^*\mathcal F}$ over ${\pi^*\mathcal F_t}$ .

Recall that two sigma-algebras ${\mathcal G}$ and ${\mathcal H}$ are relatively independent over a third ${\mathcal K\subseteq\mathcal G\cap\mathcal H}$ if

$\displaystyle {\mathbb P}(A\cap B) = {\mathbb E}\left[{\mathbb P}(A\vert\mathcal K){\mathbb P}(B\vert\mathcal K)\right]$

for all ${ A\in\mathcal G}$ and ${ B\in\mathcal H}$ . The following properties are each equivalent to this definition;

${{\mathbb E}[XY\vert\mathcal K]={\mathbb E}[X\vert\mathcal K]{\mathbb E}[Y\vert\mathcal K]}$ for all bounded ${\mathcal G}$ -measurable random variables X and ${\mathcal H}$ -measurable Y.
${{\mathbb E}[X\vert\mathcal G]={\mathbb E}[X\vert\mathcal K]}$ for all bounded ${\mathcal H}$ -measurable X.
${{\mathbb E}[X\vert\mathcal H]={\mathbb E}[X\vert\mathcal K]}$ for all bounded ${\mathcal G}$ -measurable X.

This leads us to the idea of a standard extension of filtered probability spaces.

Definition 1 An extension of filtered probability spaces

$\displaystyle \pi\colon(\Omega',\mathcal F', \{\mathcal F'_t\}_{t\ge0},{\mathbb P}')\rightarrow(\Omega,\mathcal F, \{\mathcal F_t\}_{t\ge0},{\mathbb P})$

is standard if, for each time t ≥ 0, the sigma-algebras ${\mathcal F'_t}$ and ${\pi^*\mathcal F}$ are relatively independent over ${\pi^*\mathcal F_t}$ .

Continue reading “Extending Filtered Probability Spaces” →

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. Continue reading “A Process With Hidden Drift” →

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