From the definition of standard Brownian motion B, given any positive constant c, will be normal with mean zero and variance c(t–s) for times . So, scaling the time axis of Brownian motion B to get the new process just results in another Brownian motion scaled by the factor .
This idea is easily generalized. Consider a measurable function and Brownian motion B on the filtered probability space . So, is a deterministic process, not depending on the underlying probability space . If is finite for each then the stochastic integral exists. Furthermore, X will be a Gaussian process with independent increments. For piecewise constant integrands, this results from the fact that linear combinations of joint normal variables are themselves normal. The case for arbitrary deterministic integrands follows by taking limits. Also, the Ito isometry says that has variance
So, has the same distribution as the time-changed Brownian motion .
With the help of Lévy’s characterization, these ideas can be extended to more general, non-deterministic, integrands and to stochastic time-changes. In fact, doing this leads to the startling result that all continuous local martingales are just time-changed Brownian motion.
Defining a stochastic time-change involves choosing a set of stopping times such that whenever . Then, defines a new, time-changed, filtration. Applying the same change of time to a process X results in the new time-changed process . If X is progressively measurable with respect to the original filtration then will be -adapted. The time-change is continuous if is almost surely continuous. By the properties of quadratic variations, the following describes a continuous time-change.
Theorem 1 Any continuous local martingale X with is a continuous time-change of standard Brownian motion (possibly under an enlargement of the probability space).
More precisely, there is a Brownian motion B with respect to a filtration such that, for each , is a -stopping time and .
In particular, if W is a Brownian motion and is W-integrable then the result can be applied to . As this gives
for a Brownian motion B. This allows us to interpret the integral as a Brownian motion with time run at rate .
The conclusion of Theorem 1 only holds in a possible enlargement of the probability space. To see why this condition is necessary, consider the simple example of a local martingale X which is identically equal to zero. In this case, it is possible that the underlying probability space is trivial, so does not contain any non-deterministic random variables at all. It is therefore necessary to enlarge the probability space to be able to assert the existence of at least one Brownian motion. In fact, this will be necessary whenever has nonzero probability of being finite. As stated in Theorem 1, only enlargements of the probability space are required, not of the filtration . That is, we consider a probability space and a measurable onto map preserving probabilities, so for all . Any processes on the original probability space can then be lifted to . The filtration is also lifted to . In this way, it is always possible to enlarge the probability space so that Brownian motions exist. For example, if is a probability space on which there is a Brownian motion defined, we can take , and for the enlargement, and is just the projection onto .
Theorem 1 is a special case of the following time-change result for multidimensional local martingales. A d-dimension continuous local martingale is a time change of Brownian motion if the quadratic variation is proportional to the identity matrix. Below, denotes the Kronecker delta.
Theorem 2 Let be a continuous local martingale with . Suppose, furthermore, that for some process A and all and . Then, under an enlargement of the probability space, X is a continuous time-change of standard d-dimensional Brownian motion.
More precisely, there is a d-dimensional Brownian motion B with respect to a filtration such that, for each , is a -stopping time and .
Proof: Define stopping times
and the filtration , for all . The stopped processes have quadratic variation . So, they are -bounded martingales, and the limit
for all , so Z is a martingale. From the definition, is left-continuous and A is constant with value t on the interval . Then, as the intervals of constancy of coincide with those of , it follows that , and Z is continuous.
Next, is a local martingale. As is square integrable it follows that are uniformly integrable martingales. Applying optional sampling as above and substituting in shows that
is a martingale with respect to . If it is known that , then Lévy’s characterization says that Z is a standard d-dimensional Brownian motion. More generally, enlarging the probability space if necessary, we may suppose that there exists some d-dimensional Brownian motion W independently of . Then, is a martingale under its natural filtration, with covariations . So, is a local martingale.
Then, and are local martingales under the filtration jointly generated by and M. So, by Levy’s characterization, B is a standard d-dimensional Brownian motion. For any , A is constant on the interval . So, X is also constant on this interval giving,
It only remains to be shown that is a -stopping time. For any times the definition of gives
As this holds for all u, , and is a -stopping time. ⬜
So, all continuous local martingales are continuous time changes of standard Brownian motion. The converse statement is much simpler and, in fact, the local martingale property is preserved under continuous time-changes.
Lemma 3 Let X be a local martingale and be finite stopping times such that is continuous and increasing.
Then, is a local martingale with respect to the filtration .
Proof: Choose stopping times such that the stopped processes are uniformly integrable martingales. Then, set
Continuity of gives , and as n goes to infinity. For each ,
implies that is a -stopping time. Finally, is a uniformly integrable process and optional sampling gives
So, is a local martingale with respect to . ⬜
Finally for this post, we show that stochastic integration transforms nicely under continuous time-changes. Under a time-change defined by stopping times the integral transforms as
We have to be a bit careful here, as the integral on the left hand side is defined with respect to a different filtration than on the right. The precise statement is as follows.
With respect to the filtration , is a semimartingale, is predictable and -integrable, and
Proof: First, as is continuous, changing time takes the set of -adapted and left-continuous (resp. cadlag) processes to the set of -adapted and left-continuous (resp. cadlag) processes . However, the predictable sigma algebra is generated by the adapted left-continuous processes, so the time-change takes -predictable processes to -predictable processes. Therefore, with respect to , is predictable and is a cadlag adapted processes.
If is elementary predictable, then (1) follows immediately from the explicit expression for the integral. Once it is known that is a semimartingale, dominated convergence for stochastic integration will imply that the set of bounded predictable processes for which (1) is satisfied will be closed under pointwise convergence of bounded sequences. So, by the monotone class theorem, the result holds for all bounded predictable .
To show that is a semimartingale, it is necessary to show that it is possible to define stochastic integration for bounded predictable integrands such that the explicit expression for elementary integrals and bounded convergence in probability are satisfied. We can use (1) to define the integral. Let us set
which, by the Debut theorem, will be an -stopping time and . Also, is left-continuous. Suppose that we are given a left-continuous and adapted process with respect to then, for each , will be left-continuous and adapted with respect to . Therefore
will be predictable. More generally, this holds for any -predictable . Also, , so , except on intervals for which (and hence ) is constant. So, when they are elementary predictable, equation (1) holds with these values of and .
We then use (1) to define the stochastic integral for bounded -predictable . By the dominated convergence theorem for integration with respect to X, it follows that the integral we have defined with respect to satisfies bounded convergence in probability, as required. So is indeed an -semimartingale.
Here, (1) has been applied to the bounded integrands and . Integrating with respect to both sides shows that is -integrable and as required. ⬜